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AN ELEMENTARY LOGIC 



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AN 



ELEMENTARY LOGIC 



BY 



JOHN EDWARD RUSSELL, M.A. 

MARK HOPKINS PROFESSOR OF INTELLECTUAL AND MORAL 
SCIENCE IN WILLIAMS COLLEGE 



THE MACMILLAN COMPANY 

LONDON: MACMILLAN & CO., Ltd. 
1906 

All rights reserved 



I0 



« 



'pc^\^s 



LIBRARY of CONGRESS 

Two GoDies Received 

APR 23 1906 

^ Copyright Entry 
CliASS <X XXc, No. 
' COPY B. 




Copyright, 1906, 
By the MACMILLAN COMPANY. 



Set up and electrotyped. Published April, 1906. 



J. 8. Gushing & Co. — Berwick & Smith Co. 
Norwood, Mass., U.S.A. 



PREFACE 

It is the aim of this book to aid young students in 
gaining a comprehension of the essential principles 
of correct thinking. 

It aims also to assist those teachers who find the 
use of a text-book in this subject advantageous. The 
book contains little that is original. It follows the 
main tradition of logical doctrine, with such omis- 
sions and changes in method as I, during years of 
experience, have found it desirable to make in an 
elementary course in logic. 

The more important deviations from traditional 
logic will be found in the treatment of Judgment and 
the Syllogism and in the part entitled The Logic of 
Science. I have emphasized the distinction between 
Judgment and Propositions by giving to the former 
a separate exposition, preceding the discussion of 
Propositions. 

In the treatment of the Syllogism, some technical 
matter, particularly that pertaining to moods, has 
been omitted, and changes have been made in the 
arrangement of topics which it is hoped will facilitate 
the comprehension of this subject. I hope I have 
made some improvement in the treatment of what 
is customarily called Inductive Logic, or Induction. 

5 



6 PREFACE 

I have tried to make clearer the nature of Science 
and the limits of its explanations ; and I have tried to 
give a more definite statement of the special problems 
of Science and to explain more clearly the methods 
of scientific thinking than is done in many books on 
this subject. 

I am glad to acknowledge special indebtedness, 
both in the growth of my views and in the prepara- 
tion of this book, to F. H. Bradley, Jevons, Bosan- 
quet, Sigwart, and, preeminently, to John Stuart Mill, 
whose great work is, I think, still the most valuable 
contribution to Inductive Logic that has yet been 
made. 

WiLLIAMSTOWN, 

March 19, 1906. 



CONTENTS 



CHAPTER I 

Introduction 

PAGE 

1, The Meaning of Logic. Its Relation to Knowledge . . 13 

2. Divisions in Logic. Basis of these Divisions. Consistency 

and Truth 14. 15 

PART ONE 



THE LOGIC OF CONSISTENCY 

CHAPTER II 

The Concept 

The Nature of the Concept. Concept defined and the Two 

Significations of the Term Distinguished . . 16, 17 

Kinds and Distinctions of Concepts . . . . -17 

(1) Simple and Complex Concepts . . . . 17, 18 

(2) Universal, Individual, and Collective Concepts . 18, 19 

(3) Abstract and Concrete Concepts . . . 20-22 

(4) Positive and Negative Concepts ... 22, 23 

(5) Absolute and Relative Concepts .... 23 

(6) Connotative Names, Extension and Intension of 

Concepts 24-26 



CHAPTER III 

Division, Definition, and Classification 

5. The Meaning of these Terms. Explanation of the So-called 

Predicables. Genus, Species, Differentia, Property, and 
Accident 27-29 

6. The Processes of Division, Definition, and Classification 29-34 

7 



8 CONTENTS 

PAGE 

7. Rules for Division and Definition ..... 34, 35 

8. Observations upon Division and Classification in Formal Logic 35, 36 

CHAPTER IV 

Judgments 

9. The Nature of a Logical Judgment; Analysis of Judgment 37-39 
Judgment distinguished — 

(i) From Exclamation . . . . . '. .39 

(2) From Interrogation ....... 40 

(3) From Command ....... 40 

10. Judgment and its Verbal Expression. Judgment distin- 

guished from Grammatical Sentence. Different Ways 

in which Judgment can be Expressed . . . 40-43 

11. The Kinds of Judgments — 

(1) Categorical Judgment 44 

(2) Hypothetical Judgment 44 

(3) Disjunctive Judgment ..... 49-51 
Distinctive Features of the Hypothetical Judgment. Its Re- 
lation to the Categorical Judgment. Characteristics of 

the Disjunctive Judgment. Features which distinguish 
this Judgment from the Categorical and Hypothetical 
Judgment 44-51 

12. The Quality of Judgments. The Meaning of Negation in 

Judgment. The Function of Negative Judgments 51-53 

13. Other Distinctions in Judgments ...... 54 

(1) Analytic and Synthetic Judgments .... 54 

True Character of this Distinction .... 55 

(2) Modal Judgments 55 

The Distinction between Assertorial and Apodictic 
Judgments 55. 5^ 

(3) Different Significations of " may," "can," and " must " 

in Logical Assertions 5^5^ 

CHAPTER V 

The Logic of Propositions 

14. Meaning of the Proposition. Propositions Analyzed. Logi- 

cal and Grammatical Analysis .... 59, 60 



CONTENTS 9 

PAGE 

15. Kinds of Propositions. Three Such Kinds corresponding to 

the Three Kinds of Judgments. Two Additional Sorts : 
Exclusive and Exceptive Propositions ... 61, 62 

16. Quality of Propositions. Negative Propositions; how^ Dis- 

tinguished ........ 62-65 

17. Quantity of Propositions. Quantification. Quantity in its 

Application to the Subject-term. Universal and Particu- 
lar Propositions. Quantity in its Application to the 
Predicate-term. The Method of Quantification of 
Propositions 65-73 

CHAPTER VI 
Inference. Reasoning 

18. The Nature of Inference. Inference Defined. Its Essential 

Elements. The Criteria or Tests of Inference. Infer- 
ence distinguished from Other Mental Operations. 
Views of Mill and Other Logicians Criticised , 74~^S 

19. The Forms of Inference. Deductive and Inductive Inference 

Distinguished. Varieties of Deductive and Inductive 
Inference 85-89 

CHAPTER VII 
The Forms and Methods of Deductive Inference 

20. EquipoUence : Its Meaning and the Two Relations on which 

it is Based. Various Ways in which EquipoUence is 
Maintained. Obversion, Conversion, Contraposition, 
and Added Deterniinants 90-96 

21. Inference by Opposition. Contrary, Contradictory, Sub- 

contrary, and Subaltern Propositions . . . 96-99 

CHAPTER VIII 
Mediate Inference. The Syllogism 

22. Description of the Syllogism. The Categorical, the Hypo- 

thetical, and the Disjunctive Syllogism. Irregular Syllo- 
gism. Dilemma. Enthymeme. Sorites . . 101-I15 



lO CONTENTS 



23. Regulative Principles and Rules for the Syllogism. Rule for 

the Disjunctive Syllogism, Basis of the Hypothetical 
Syllogism and the Consequent Rules for its Valid Use. 
Rules for the Categorical Syllogism ascertained and 
systematically Stated ...... 1 15-126 

CHAPTER IX 

Fallacies in Deductive Reasoning 

24. Description of Fallacies. Explanation of them and the Main 

Division of them into Formal and Material Fallacies. 
Specific Exposition of Material Fallacies. Formal Fal- 
lacies Explained ....... 127-138 

25. Classification and Technical Designation of Fallacies . 138-142 

26. Value of the Syllogism 142-150 

PART TWO 

THE LOGIC OF SCIENCE 
CHAPTER X 

Introductory 

27. The Meaning of Science. The Scientific Conception of the 

World. Scientific Explanation. The Special Problems 
of Science Explained. Ascertainment of Causal Con- 
nection by Observation and Experiment. Explanation 
by Hypothesis. Calculation of Chances and the Method 
of Statistics 152-159 

CHAPTER XI 

The Ascertainment of Causal Connection by 
Observation and Experiment 

28. Observation and Experiment. Importance and the Difficul- 

ties of Observation. Advantages of Experiment. The Aid 

it affords in the Ascertainment of Causal Connection 160-165 

29. The Regulative Principles for Observation and Experiment. 

The So-called Inductive Methods. The Aim of these 
Methods illustrated by their Use .... 165-174 



CONTENTS 1 1 

PAGE 

30. The Logical Value of the Methods of Observation and 

Experiment 175-180 

CHAPTER XII 

Explanation by Hypothesis 

31. The Essential Features of a Scientific Hypothesis. The 

Requisites of a Legitimate Hypothesis . . . 182-184 

32. The Method of Explanation by Hypothesis. The Two Essen- 

tial Processes. Construction of Hypothesis and Verifi- 
cation. Complete Verification and Proof distinguished 
from Incomplete Verification and Disproof . . 184-19 1 

33. The Value of Rejected Hypotheses .... 191, 192 

CHAPTER XIII 
The Third Special Problem in the Logic of Science 

34. Calculation of Chances. The Meaning of Chance and Prob- 

ability. The Method of calculating Chances. The Theory 

of Probability 193-203 

35. The Method of Statistics and the Uses of this Method . 203-207 

CHAPTER XIV 

Generalization from Experience and Analogy 

36. Inductive Generalization and its Varieties. Analogical Infer- 

ence. Its Scientific and its Practical Value. Principles 
Vi'hich regulate its Employment .... 208-2H 

CHAPTER XV 

Fallacies 

37. Fallacies Incident to reasoning upon Matters of Fact. Four 

Groups of these Fallacies : Fallacies in Explanation by 
Observation and Experiment. Fallacies in Explanation 
by Hypothesis. Fallacies Incident to the Calculation 
of Chances and the Method of Statistics. Fallacies of 
Generalization and Analogy . . . . 212-223 

APPENDIX. Practical Exercises and Questions . . 224 



CHAPTER I 

INTRODUCTION 

Section i 

the meaning of logic 

Logic is the science of thought. Its aim is to ascer- 
tain and apply the principles which are regulative for 
right thinking. By thinking in logic is meant the fa- 
miliar mental operations of forming ideas, or notions, 
making assertions, and reasoning. By right or vahd 
thinking is meant that mode of thinking which attains 
its end, which is consistency, truth, and knowledge. 
The subject-matter of logic being thought, this science 
is not primarily concerned with things or facts of expe- 
rience ; it is concerned with the knowledge of facts only 
so far as that knowledge is attained by thinking. 

The aim of all serious thought is knowledge; and 
logic aids the attainment of knowledge in two ways: 

(i) It affords a negative criterion of truth or test of 
knowledge ; for whatever violates the laws of thought 
cannot be true in any real world. 

(2) Logic aids knowledge by defining the principles 
and laws of thought in accordance with which all 
knowledge beyond mere sense perception is attained. 

13 



14 ELEMENTARY LOGIC 

If I am to reach knowledge, my thinking must con- 
form to certain principles and laws ; and it is the func- 
tion of logic, as we have said, to ascertain and apply 
these constructive principles. 

Section 2 
divisions in logic 

The division of logical doctrines should be based upon 
a difference between two aims of thought, rather than 
upon the two forms of inference known as deductive 
and inductive. 

These two kinds of inference are associated with 
these two distinct fields of logic, but the difference 
between them is not the basis upon which a division of 
logical doctrine should be made. Logical thinking 
aims at two things: consistency and knowledge of 
fact. By consistency we mean that connection between 
a given judgment and other judgments which makes 
a judgment true if the given one is true, and false if the 
given one is not true. 

Consistency means, if I think and assert this, I must 
also assert that. By truth in matters of fact, I mean 
agreement or correspondence between thought and fact ; 
and the certainty of this agreement is knowledge. This 
difference in the aim of our thinking is the true princi- 
ple of a division of logic. In accordance with it we 
have the Logic of Consistency and the Logic of Sci- 
ence. The former is customarily called Formal or 
Deductive Logic ; the latter, Inductive Logic. 



INTRODUCTION 1 5 

The function of Formal or Deductive Logic being to 
establish consistency in our thinking, the function of 
Inductive Logic the attainment of knowledge, I accord- 
ingly divide this study into two parts : — 

Part One will deal with the Logic of Consistency; 
Part Two, with the Logic of Science. 



PART ONE 

THE LOGIC OF CONSISTENCY 

CHAPTER II 

THE CONCEPT 

Section 3 

the nature of the concept 

In formal logic it is customary to present the doc- 
trine of thought under three principal topics : The Con- 
cept or Names or Terms, Judgments or Propositions, 
and Inference. 

I shall adhere to this division and order of topic, and 
proceed first to the Concept. 

DEFINITION 

A Concept is a mode of thinking in which something 
is by our thought treated as one thing and distinguished 
from all other things. In the objective sense of the 
term, a concept is anything which can be thus thought 
as one thing, identical with itself and different from 
other things ; unity and distinctness from other things 
are consequently the defining characters of the concept 
in the objective meaning of the term. 

16 



THE CONCEFr 1 7 

The subject-matter, or content, of a concept is its 
meaning. The features by which this meaning is dis- 
tinguished from other meanings constitute the marks, 
properties, or attributes of a concept. The sum total 
of these defining marks which are essential to the mean- 
ing of a concept are what is meant by its intension, or 
connotation. 

I have chosen the word Concept in preference to 
Name or Term, which are more commonly used, be- 
cause concept makes more prominent the mental act 
or logical process that is being studied; term or name 
designates properly that which is the product or result 
of this act. It is regrettable that there is no good word 
for naming this mode of thinking. Concept in present 
usage is ambiguous ; it is not always easy to determine 
whether it is the mental operation itself, or the thing 
thought about that is meant. The words subjective 
and objective are the best we have to mark this differ- 
ence in the meaning of the term ; concept in the sub- 
jective sense being the mental act, and concept in the 
objective sense meaning the product of this act or that 
which is meant. It is in this objective sense I shall 
use the term unless I give notice of my intention to use 
it in the other signification. 

Section 4 

the kinds and distinctions of concepts 

I. Simple and Complex Concepts. — The content of 
a concept can be extremely simple, and its range very 



1 8 ELEMENTARY LOGIC 

limited ; it can be something that is incapable of analy- 
sis or description, as, for example, this particular shade 
of blue I observe in the sky. This is a concept if, in 
addition to its being my momentary perception, I think 
it ; if I judge respecting it that it is Hke the blue I saw 
in the waters of Lake Geneva ; for by so doing I recog- 
nize this bit of color as one thing, distinguishable from 
other things. The content of a concept may be ex- 
tremely complex, and its range of meaning illimitable. 
The universe itself may be a concept. Accordingly, 
considered in respect to their structure, concepts are 
Simple or Complex. 

Simple concepts are those which do not admit of 
analysis or separation into other concepts. These 
concepts are formed at a stage of mental development 
which precedes the more conscious and purposeful 
thinking with which logic deals. 

Complex concepts are those which can be resolved 
into other concepts. These concepts are the products 
of dehberate thinking. Such are the concepts of 
Science, Philosophy, and those for the most part which 
are employed in our ordinary intercourse with each 
other. 

2. Concepts are again distinguished as Universal, 
Individual, and Collective. 

A universal concept is formed by uniting in thought 
those features or marks and those only which are 
common to a number of individuals. 

An individual concept is one which is formed by 
uniting in thought those marks which distinguish this 



THE CONCEPT I9 

individual from all other individuals. In traditional 
logic a general name, answering to a universal 
concept, is a name which, in the same signification, 
is applied to all the individuals which constitute a given 
class or totality. A singular name is one which in the 
same signification can be applied to but one individual. 
A collective concept is one which is formed by uniting 
in thought those marks or properties in virtue of which 
a number of individuals are considered, not separately 
or distributively, but as forming a single body or or- 
ganism. For instance, in forming the concept army, 
I consider a number of individual men only so far as 
they are taken together and united in a particular form 
of organization; they thus form a single body. The 
individual men who compose an army are not considered 
distributively, but collectively; this collection implying 
both plurality and unity. Hence, this concept par- 
takes of the character of both the universal and the 
singular concept. The collective name of traditional 
logic, hke the general name, implies a number of in- 
dividuals; but, unhke the general name, the collective 
name cannot be applied to these individuals taken 
distributively, but only as they are taken together. 
The name army, for instance, is not applicable to the 
individual soldiers which constitute it; the name 
horse is applicable to individual animals; but this 
name army is applicable only to that single organiza- 
tion which these individual soldiers all taken together 
constitute. The name army apphes only to this unity 
of all the individuals so united. 



20 ELEMENTARY LOGIC 

3. Abstract and Concrete Concepts. — According to 
the point of view from which their subject-matter 
is regarded, concepts are Abstract or Concrete. The 
true distinction between these is best apprehended 
by observing two things in the formation of concepts. 

The first is, that in forming any concept, particularly 
in forming the general concept, some abstraction 
is always involved, since this concept is formed by 
uniting only the marks which are common to a num- 
ber of individuals, and there is consequently an ab- 
stracting from the other marks which belong as truly 
to these individuals as do the marks that are included 
in the concept. The difference between the abstract 
and the concrete concept does not he in the different 
ways in which these concepts are formed; not in the 
fact that abstraction takes place in the formation of one 
concept and not in the formation of the other; for 
some amount of abstraction is involved in the formation 
of any concept. 

The second thing to observe is that there are two 
ways of treating the subject-matter of any concept ; we 
may either consider this subject-matter as something 
which possesses properties or marks, in which case 
we can consider these properties or marks only as 
they are conceived as belonging to this subject- 
matter; or, we may single out one or more of these 
properties, and by abstracting them, so to speak, from 
that to which they belong, form another concept. For 
example, in the concept of a Centaur, the various marks 
which constitute this concept, say a, b, c, d, . . . x, are 



THE CONCEPT 21 

considered only as they belong to the something called 
Centaur; and this something is thought of only as pos- 
sessing these properties. Now let us select one of these 
marks, say b, and, by abstracting from the other marks, 
we form a concept, — the concept of the particular 
property designated by this letter b; this concept is 
abstract, not because there has been abstraction in 
forming it, but because its subject-matter is considered 
apart from the Centaur of which it is a property. 

Accordingly a concrete concept is to be defined as 
one in which the subject-matter is considered as some- 
thing which has properties, and its properties are con- 
sidered only as they belong to this subject-matter. 

An abstract concept is to be defined as one in which 
the subject-matter, while it can be the property of some- 
thing, is considered in abstraction from that something 
to which this property would belong; thus, whiteness 
is the property of some conceivable thing, but this prop- 
erty is considered apart from any subject-matter to 
which it can be attached ; and when it is so considered, 
the concept is abstract. 

This view of the distinction between concrete and 
abstract concepts explains the fact that many of the 
abstract names in logic are ambiguous, it being impos- 
sible to determine in the lists of such names given in 
text-books which are abstract and which are concrete. 
These distinctions as we have explained them are not 
fixed ; some words doubtless are always names of con- 
crete concepts, as horse, animal, man, state ; others are 
perhaps as uniformly names of abstract concepts; but 



22 ELEMENTARY LOGIC 

very many such words are in some situations names of 
abstract concepts and in other connections they are 
names of concrete concepts. For instance, in most 
situations charity would be rightly called an abstract 
name or concept; but in the passage, "Charity suffer- 
eth long and is kind," the concept is concrete; it is so 
because it is considered and treated as a something 
which has properties ; here the attributes are long-suffer- 
ing and kind. 

The same name can therefore be concrete in one 
usage and abstract in another; only the known inten- 
tion and use of the author can determine in such 
cases whether the concept is abstract or concrete. 
Thus, in the proposition, "The mercy of the Lord 
is from everlasting to everlasting," mercy is an ab- 
stract concept; but in the sentence, "Mercy and peace 
shall go before him," the same name is concrete. 

4. Positive and Negative Concepts. — According as 
the subject-matter implies affirmation or negation, 
presence or absence of this subject-matter, concepts 
are Positive or Negative. What it is to affirm or deny 
will be explained in the chapter on Judgments. The 
signs of negation are easy enough to recognize, and 
there should be no difficulty in distinguishing be- 
tween positive and negative names. Such prefixes as 
not, non, invariably denote negative concepts. 

Logicians are wont to distinguish a class of concepts 
which they call Privative. These, it is said, denote the 
mere absence of some state or quahty which naturally or 
normally belongs to a subject ; for example, blindness , 



THE CONCEPT 23 

deafness. These terms, it is said, have no meaning if 
applied to beings which do not normally possess the 
powers of sight and hearing. On the other hand, the 
terms not-seeing, not-hearing, are negative; they are 
appUcable to inanimate beings as well as to beings which 
can possess such powers. The difference between 
negative and privative is, however, one of degree only; 
a privative name is a negative name which has the 
additional imphcation that the subject ought nor- 
mally to possess the specified mark or property. 

5. Absolute and Relative Concepts, — In formal 
logic a name is called Absolute when its meaning is 
complete without involving a relation to another 
name. A name is Relative if its complete connota- 
tion does involve a relation to some other name. For 
example, the names husband, parent, brother, king, 
are relative names; while the names, metal, dog, man, 
happiness, are absolute names; the meaning of hus- 
band, parent, brother, king, is not complete without 
the relation to wife, other brothers or sisters, subjects; 
while the meaning of metal, dog, man, happiness, in- 
volves no such necessary relation. 

The meaning of a relative name must not be con- 
founded with the meaning which a name may have for 
any mind in consequence of all it suggests to that 
mind ; in the logical sense of the term a name is not 
relative because it suggests various other names, every 
name does that; but because that which is suggested 
is a part of its meaning. The name home suggests to 
my mind a thousand things, no one of which is any 



24 ELEMENTARY LOGIC 

part of the logical connotation of this name; likewise 
the name father calls to my mind a hundred things 
which are not part of the logical meaning of this 
name; but this name does suggest one thing which is 
a necessary part of its connotation, viz. offspring, son 
or daughter ;• and apart from that relation and those 
two related things, this name has no logical meaning. 

6. Connotative and Denotative Names, Extension 
and Intension of Concepts. — By some logicians names 
are distinguished as Connotative and Denotative. A 
connotative name is one which both designates a 
thing and imphes attributes of this thing; man, for 
example, is a connotative name, since it means indi- 
vidual beings and implies certain attributes belong- 
ing to these individuals. A denotative name is one 
which points out or distinguishes some individual, 
and does not imply in its meaning attributes. 

I have given the customary distinction ; but it is not, 
I think, the right one. The real distinction is that of 
function or use of the name, or rather the intention 
of the thinker. In using the name man, the purpose of 
the thinker is to designate a certain group of qualities 
or attributes which are possessed in common by an 
indefinite number of individuals; these individuals are 
of importance, and are meant, only so far as they possess 
these attributes; it is the attributes and not the indi- 
viduals that the name signifies, which constitute its 
connotation. The name John Smith is used to point 
out this individual, and to distinguish him from other 
individuals. 



THE CONCEPT 2$ 

This distinction of connotation and denotation is not, 
properly speaking, a distinction in names themselves; 
it is rather a distinction in the purpose or intention of 
the one who is dealing with that which the names 
mean. The distinction which these logicians make is 
identical with that between universal or general, and 
individual or singular names. 

We have seen that every name has a connotation; 
there are no denotative names in the sense in which these 
logicians use the term denotative, viz. a name which 
connotes no attributes. Even such individual names as 
John Smith must have some connotation ; for it is only 
by virtue of some properties that I can distinguish this 
individual named John Smith, my John Smith, from 
other individuals, or from the John Smith that somebody 
else may mean. 

The distinctions of Extension and Intension apply to 
the subject-matter of every concept, and it is better to 
make extension synonymous with denotation and inten- 
sion synonymous with connotation. Accordingly, by 
the intension or connotation of a concept or name is 
meant the marks which are common to all the individ- 
uals to which the name appHes; and by the extension 
or denotation of a concept or name is meant the indi- 
viduals which possess these marks. 

Every name has accordingly both extension and in- 
tension; this extension may be very great in the cases 
of names of classes or it may be reduced to a single 
individual in the case of a singular name. So with in- 
tension : it ranges from very few marks to an innumer- 



26 ELEMENTARY LOGIC 

able number of marks. A law of relation between 
extension and intension is sometimes stated in this 
way; extension and intension vary in inverse ratio, or 
the greater the extension of a name the less is its in- 
tension. A glance at the way in which general con- 
cepts are formed makes^ the truth of this statement 
obvious. 



CHAPTER III 

DIVISION, DEFINITION, AND CLASSIFICATION 

Section 5 

the meaning of these terms 

By Division in logic is meant the separation of a given 
class into the lesser classes which are contained within 
it. By Definition is meant the specification or state- 
ment of the marks which distinguish these lesser classes 
within some larger containing class. By Classification 
is meant the systematic arrangement and distribution 
of individual objects so as to form classes. 

Division and classification are complementary pro- 
cesses; and these processes cannot in reahty be sepa- 
rated from each other ; every division involves a classi- 
fication, and every classification is at the same time a 
process of division. 

A very intimate relation also exists between division 
and definition. A definition is involved in every process 
of division; and division is possible only as there is 
at the same time definition. I cannot systematically 
and completely divide a class without at the same 
time defining each of the successively formed classes; 

27 



28 ELEMENTARY LOGIC 

nor can I define any one of these classes without at the 
same time dividing the larger class in which it is con- 
tained. It should also be observed that division and 
definition are closely related to extension and intension 
of names, division being the systematic statement or 
unfolding of the extension of a name, definition the 
systematic statement of the intension of a name. 

Before proceeding to the exposition of these processes 
I have defined, it is necessary to explain a group of 
terms which occur in formal logic; they are the so- 
called predicables. Genus, Species, Differentia, Prop- 
erty, and Accident. 

A genus is the larger class in the process of logical 
division; species are the lesser classes into which the 
genus is divided. Difference or differentia are the 
marks which distinguish the species from each other 
within a given genus. Property or proprium is a 
mark which belongs to every individual of a class, but 
which is not a part of the connotation or meaning 
of the name ; thus, equahty of its three angles to two 
right angles is a property of every triangle, but this 
mark is not essential to the meaning of a triangle, not 
necessary to its logical definition. By accident is meant 
a mark which belongs to some individual or to a part 
of the individual members of a class, but never to all 
the members of a class ; for example, a diameter of six 
inches may be a mark of a certain circle, or of a num- 
ber of circles; but it is not a mark of all circles. 

There is a twofold distinction between property and 
accident : — 



DIVISION, DEFINITION, AND CLASSIFICATION 29 

(i) A property is supposed to depend upon the 
essential nature or essence of the species or genus; 
while the accident does not. Thus, the equality of the 
radii of a circle depends upon the essential nature of 
the circle; hence it is a property of all circles; but 
a given length of a radius, say six feet, does not depend 
upon the nature of all circles; it is only an accident 
of certain circles or of one circle. 

(2) The property mark must belong to every in- 
dividual of the class; the accident cannot belong 
to every individual, and may belong to only one in- 
dividual. The student must not understand that genus 
and species in logic designate fixed classes ; they are, 
on the contrary, only relative distinctions. The same 
class can be a species relative to a larger class of vi^hich 
it is a part, and a genus relative to a lesser class into 
which it can be subdivided; thus, horse is a species 
relative to the genus quadrupeds, and a genus to the 
various kinds of horses into which we can divide this 
class. 

Section 6 

the processes of division, definition, and classi- 
fication 

From this explanation of the technical terms, we pass 
to the processes of division, definition, etc. We can 
best understand these processes if we study a concrete 
case. Accordingly, let us suppose we are to divide the 
books in a hbrary. Our problem is both one of division 
and classification; for, at the outset, these books are a 



30 ELEMENTARY LOGIC 

class of things in virtue of certain marks common to 
them all. Now, it is obvious that different divisions 
and different classifications of these books are possible. 
The first step, therefore, in our task is the selection of a 
principle, or basis, of this proposed division and classi- 
fication ; in technical phrase a fundamentum divisionis. 
We will accordingly make the subject-matter of which 
these books treat the basis of the division, the funda- 
mentum divisionis. We first divide the genus books 
into the following species ; Histories, Books of Science, 
Literature, and Philosophy. This division is neither 
exact nor exhaustive, but it will answer our purpose. 
We note that the species are coordinate; the follow- 
ing diagram will show this : — 

BOOKS 

History Science Literature Philosophy 

We note also that a peculiarity of the subject-matter 
treated in each of these species of books constitutes the 
differentia of each species: thus, the differentia of the 
history books is, that they treat of the actions of human 
beings in society. 

Now let our division be carried one step farther, and 
we obtain the following new classes: History gives 
Ancient, Mediaeval, and Modern History; Physical 
Science gives Physics, Chemistry, and Biology ; Litera- 
ture gives Fiction, Essays, and Poetry; Philosophy 
gives Logic, Theory of Knowledge, and Metaphysics. 
There results from this second step in our division the 
following things : — 



DIVISION, DEFINITION, AND CLASSIFICATION 3 1 

(i) The lesser classes, which were species in the 
first step in division, now become genera in relation 
to the classes into which they have been separated; 
for example. Histories are a species of Books; but 
Ancient Histories are a species of histories, that is. 
Histories are both species and genus, species relative to 
the class Books, and genus relative to the class An- 
cient Histories. 

(2) We note that a change takes place in the basis 
of division, or the jundanientum divisionis, in this 
second stage of the division. For the histories the basis 
of division becomes the period of time to which the 
historical phenomena belong ; for the books in Physical 
Science, it is the special groups of natural phenom- 
ena which science comprehends. 

(3) This process of division explains some distinc- 
tions which the student will find in text-books on 
logic; these distinctions are, summum genus, infima, 
and proximate species. The summum genus is the 
largest class with which a process of division begins, 
the largest class therefore in a system of division, or in 
a scheme of classification ; the infima species is the class 
in which a complete division or any division terminates ; 
thus, were I to complete this division of books, the 
last class of books would be infima species. By a proxi- 
mate genus is meant the class that in the scale of divi- 
sion is nearest to the classes below it ; and it should be 
added that these lesser classes are the proximate species 
of this genus ; thus, in the division made above. Litera- 
ture is the proximate genus to Fiction, Poetry, etc., but 



32 ELEMENTARY LOGIC 

not the proximate genus to English Poetry, or German 
Poetry. Had we subdivided Hterature into Enghsh 
Poetry, German Poetry, etc., we should have gone 
beyond or passed over the proximate species of literature. 

Let us now return to the starting point in the division 
of these books, and make use of a different method or 
principle of division, a different sort of jundamentum 
divisionis. 

Let the basis of division now be the possession or 
non possession of a given mark ; and our division will 
be the following: Books are divided into Histories 
and non historical books. Histories into Modern His- 
tories and those that are not modern; non historical 
books are divided into Literature and those which are 
not books on hterature, etc. This method of division is 
technically called division by dichotomy, or dichoto- 
mous division, since it makes two classes only at each 
step of the process. This method of division has the 
advantage of being exhaustive at each step; since the 
class that has the negative mark includes all the indi- 
viduals that are not put into the class which has the 
positive mark. 

Let us turn now to the other closely allied process, 
definition; and, using the same case, the division of 
the books in the library, the problem now is to deter- 
mine the marks or characteristics which distinguish 
each species of books. These marks define or bound 
off each of the several species into which the genus books 
was divided. If we examine the individual books 
which constitute only one of these species, say Histories, 



DIVISION, DEFINITION, AND CLASSIFICATION 33 

we shall see that two sets of marks belong to them; 
the marks which make them members of the genus 
Books, and the marks which make them members of the 
species Histories; in other words, each history book 
possesses generic and specific marks. 

A statement of both these marks is what the logicians 
mean by a definition: thus. Histories are defined as 
books which treat of the actions of men in society; and, 
since these specific marks are at the same time differ- 
entia, a definition consists in a statement of genus, 
species, and differentia. It must be kept in mind that 
it is only the generic and specific marks that are to be 
included in a logical definition. The property and the 
accident marks must never be given in such definitions. 
Definitions by property and especially by accident are 
serviceable for practical purposes, and are sometimes 
the only definitions possible ; children for the most part 
define by such marks only ; but such definitions are not 
admissible according to the canons of logic. 

Another thing should be noted : the logical definition 
applies to a class of individuals, and not to any one in- 
dividual as such; hence, only general names can be 
logically defined. The only way in which the individual 
can be defined is by a statement of all the marks which 
this individual alone possesses, and which consequently 
distinguish it from all other individuals ; thus, if I were 
to define King Edward the Seventh, I should need to 
state every mark which this man alone possesses by 
which he is different from every other being in the 
universe. If I define King Edward by saying he is an 



34 ELEMENTARY LOGIC 

Englishman, etc., I state only those marks which he has 
in common with other men of this nationality ; I define 
the species to which he belongs, not King Edward him- 
self. Of course, we always do define to some extent 
individual or singular names, we must do so in order to 
distinguish individuals ; but this process is not what the 
logicians mean by definition. 

Section 7 
rules for division and definition 

A number of such rules is usually given by logicians. 
The exposition of these processes I have given renders 
most of the rules laid down in text-books needless. It 
is an obvious corollary from the principle of division, 
that but one fundamentum divisionis can be employed 
in the same stage of the division ; otherwise confusion 
of the species or cross divisions will result. Forinstance, 
had we divided books into Histories, books on Physical 
Science, and, say, octavos, we should have used two bases 
of division at the same time; and the consequence 
would have been a confusion of the subclasses; the 
species would have overlapped ; there would have been 
cross divisions. Hence, a first regulative principle in 
division is, the fundamentum divisionis must not be 
changed during a given stage of the division. 

Again, let me suppose that, in the division of those 
books, we had in place of books on Physical Science put 
Chemistries ; by so doing we should have put a subor- 
dinate species in the place of a coordinate one; we 



DIVISION, DEFINITION, AND CLASSIFICATION 35 

should have omitted a proximate species, and the con- 
sequence would have been that some of the books in 
our hbrary would not have been divided or classified. 
Hence, a second regulative principle in division is, all\ 
the species in a given genus must be coordinate, or the 
genus must always be divided into its proximate species. 
It is obvious that a logical division should be exhaustive 
so far as it goes ; that is, all the species which belong 
to a genus should be given; the division need not be 
complete in the sense of being continued until classes 
are reached which cannot be subdivided, but the divi- 
sion should exhaust each genus that is formed in the 
process of division. The student must not confound 
exhaustive division with complete division. It is no 
logical requirement that a division be completed ; but 
it is a requirement that it be exhaustive. 

For definition it is hardly necessary to give a special 
rule. A correct definition is, as we have seen, one which 
states all the generic and the specific marks, and only 
those marks. A definition which includes an accident 
mark is too narrow, and one which contains property 
marks also contains what is superfluous. 

Section 8 

some observations upon division and classi- 
fication in formal logic 

I. It is not unimportant to distinguish between the 
classifications of formal logic and scientific classifica- 
tions. The aim of a scientific classification is to state 



36 ELEMENTARY LOGIC 

the actual relations of facts and phenomena in nature ; 
and a classification is true, only if it conforms to the 
nature of those beings and phenomena; the classifica- 
tions of logic are abstract, and, in relation to matters 
of fact, hypothetical. It is not necessary that a logical 
classification should agree with the objects and rela- 
tions in rerum natura; it is only necessary that such a 
classification shall inwardly be coherent and consistent : 
a classification is conceivable which should be logical 
throughout, but which should agree with a scientific 
classification in no other circumstance save in being 
self-consistent and coherent. A scientific classification 
must, in order to be true, conform to logical principles ; 
but a classification can be logical without at the same 
time being true. Classifications, therefore, which mod- 
ern science rejects because they are untrue, are not less 
logical in their structure than are the classifications 
which science has put in their places. 

2. The function of logic being regulative only, it 
does not teach us what are the properties and relations 
of things on the basis of which the classifications of 
science are made; it only teaches us how to construct 
a classification when we have found our things and 
classifying principles. Hence, the difficulties we en- 
counter in our attempts at scientific classification are 
only in a small degree logical ones. They arise for the 
most part from our imperfect knowledge of the things 
themselves ; and formal logic affords no remedy for this 
ignorance ; since logic is not an organum of knowledge, 
but a doctrine of thought. 



CHAPTER IV 

JUDGMENTS 

Section 9 

the nature of a logical judgment 

Judgment is the mental act of perceiving and assert- 
ing a relation between two distinguishable things. 
There are two distinct operations in an act of judgment : 

(i) the operation by which two things are distin- 
guished and related to each other; 

(2) the mental assertion that this relation thus recog- 
nized is a fact in the real world. 

As the judgments we make relate to all sorts of mat- 
ters, — things which exist in the sensible world, things 
that exist only in imagination, things perceived by sense, 
and things thought only, — so the two things which in 
judging are distinguished and related can be of the most 
various descriptions, a shade of color in a rose, the rose 
itself in the garden, the little flower plucked from the 
crannied wall, the universe of which that flower is a 
part, some action of an actual human being, some deed 
of a character in fiction, mathematical entities, the sym- 
bols of pure logic, ^ , 5, . . . X, etc. We judge about all 

37 



38 ELEMENTARY LOGIC 

such matters ; and whenever we do so, these two things 
constitute the operation itself: — 

(i) a relation between two things is perceived, ap- 
prehended ; 

(2) we entertain this relation with the conviction of 
truth, — in other words, our minds assert this relation; 
for, to have the conviction of truth and to assert, are 
one and the same thing. 

To make this clearer, let us take some cases of 
judgment : I see a flower, and I judge concerning it that 
the color of its petals is deep crimson. Let us analyze 
this act of judgment. First, there is before me a certain 
object, a portion of the real world present to my senses; 
secondly, my thought notes and distinguishes in this 
total object or presentation, two things, — that which I 
have previously learned to be the petals of this rose, and 
a feature of these petals, the color of these petals, deep 
crimson; thirdly, I perceive a relation between these 
two distinguished things, rose petals and deep crimson 
color, — this relation is that of identity, the feature color 
I perceive in the petals of this rose is identical with that 
which I know as deep crimson; fourthly, I assert that 
this relation so present to my thought is a fact or is 
true. This which I call assertion as a mental act is 
just that conviction of truth or of reality of which I am 
conscious as I attentively consider this rose. 

Again, suppose I am watching two boats, A and B, 
on a river; and I judge boat A is farther up the stream 
than boat B. Here, to my judging thought two objects 
are present, occupying a region of space in the real 



JUDGMENTS 39 

world of sense perception ; my thought distinguishes a 
definite relation between these two objects; that relation 
is in this case one of position in space. My mind as- 
serts this relation ; that is, it has the conviction that this 
relation between these two boats exists here and now in 
the real world. 

Once more, let the case be that of the poet who 
said, "Truth crushed to earth shall rise again." The 
matters here do not belong to the world of percep- 
tion ; the judgment is about the abstract things of truth 
and victory over seeming defeat, matters of character 
and spiritual experience; but the same two essential 
thought operations are here discoverable ; two things 
are distinguished and related — a condition of truth, 
viz. crushed to earth, and another conceived condition 
of the same truth, viz. rising again, victorious over seem- 
ing defeat. The relation between these two things we 
will call one of time, — the rising of truth from the 
earth is to follow her being crushed to earth. This re- 
lation is asserted in the mind of the poet ; he feels the 
conviction of truth when he says these words. 

These examples of judgment are sufficient to make 
clear, I trust, the nature of the mental operations that 
constitute it. The exact meaning of this all-important 
function will become more definite if we distinguish 
judgment from certain other mental acts and states. 

(i) From exclamation. In pure exclamation there is 
no assertion. In place of assertion, there is a state of 
feeling, emotion, such as wonder, joy, sorrow, etc. 
When I exclaim Oh ! Alas ! Ah ! and do no more, I have 



40 ELEMENTARY LOGIC 

not judged; I have not entertained anything as real; 
I have made no assertion about the real world ; I have 
simply uttered a state of feeling. 

(2) From question. A question is a mental state 
in which something is merely suggested, merely pre- 
sented to thought, but not asserted. The conviction 
of reality is that which distinguishes a judgment from 
a question. So long as I must put an interrogation 
mark after my thought, I do not judge. 

(3) From command. The essence of a command 
is the expression of will that something now conceived 
as possible become a fact; the essence of assertion 
is that something is now fact. 

The distinctive feature of judgment, its differentia, 
seems to be this conviction of truth or of reality, 
of which any one can be conscious when that one 
judges. The subject-matter can be the same, whether 
I judge or exclaim or question, or command ; the sole 
difference hes in my mental attitude to this subject- 
matter. When I judge, I claim for this subject-matter 
reality, a place in the real world ; in the other mental 
states, I do not make such a claim for that about which 
I may be thinking. 

Section 10 

judgment and its verbal expression 

I . Judgment is a mental act which can take place only 
in the mind of some individual thinker ; words are the 
signs to other minds that this act has taken place. It 



JUDGMENTS 4 1 

is customary in logic to regard the sentence which ex- 
presses a judgment and the judgment as identical 
things. It is to be admitted that they are most inti- 
mately related, as thought and speech necessarily are; 
but to identify the mental judgment with the sentence 
which expresses it leads to misapprehensions. 

One such error is the assumption that a judgment 
consists of parts as does the sentence ; and that, hke the 
sentence, it is formed by the combination of these parts ; 
and consequently it is assumed that grammatical 
analysis and analysis of the judgment are the same 
thing. Now this conception of the judgment and 
its relation to a grammatical sentence is erroneous. 
To make in the case of a judgment the same divi- 
sion into parts which is made in the sentence is seri- 
ously to misapprehend the nature of judgment. The 
judgment does not consist of parts or elements as does 
a grammatical sentence ; the judgment does not have a 
subject in the sense in which the sentence has a subject. 

In the sentence, the subject is that about which 
something is asserted by the predicate; and this 
subject is a separable part of the sentence. In a 
judgment, if we speak of a subject at all, it means the 
entire subject-matter with which the judgment deals; 
for example, in the sentence, "The stars shine," the 
subject is stars; but the subject-matter of the judg- 
ment expressed in this sentence is that portion of the 
real world of sense perception which is named by all 
the words which compose this sentence. Thus, the sub- 
ject or the subject-matter of a judgment includes both 



42 ELEMENTARY LOGIC 

subject and predicate of the grammatical sentence in 
which that judgment is expressed. 

Again, in a sentence the predicate is that which 
is affirmed or denied of the subject. In a judgment 
assertion does not consist of affirming or denying one 
thing of another thing. In a judgment there is no 
such thing as a predicate in the grammatical meaning 
of that term. When I judge "Waters on a starry night 
are beautiful and fair," I do not assert one thing about 
another thing, viz. beautiful and fair about waters; 
what I mentally do is to assert that the relation of 
identity between what I mean by waters-on-a-starry- 
night and beautiful and fair is a fact in the real world. 
Grammatical predication and logical assertion, there- 
fore, are different things. 

2. What words and what sentences can express a judg- 
ment ? The doctrine generally held is, that it is the 
declarative sentence only which can express a judg- 
ment. It is perhaps true that this sentence is the only 
way in which a judgment can be expressed without a 
possible doubt or ambiguity; but it is not true that in 
usage no other sentences do convey clearly, and with 
practical certainty, the fact that one has judged. In- 
deed, single words discharge this function ; and, at the 
earlier stages of mental development, the single word is 
the only expression of judgment. When the child says 
"hot, burn, hurt," it has as unmistakably expressed a 
judgment, "This is hot, it burns, I am hurt," as if it 
had used a complete sentence. 

The exclamatory sentence, though its primary func- 



JUDGMENTS 43 

tion is to express feeling, does, nevertheless, make 
others certain that the exclaimer has judged as well as 
exclaimed. When one exclaims, " Oh, what a beautiful 
day ! " there is as httle doubt in the minds of those who 
hear, that a judgment has been uttered, as there is 
that an emotion has been expressed. The truth is, no 
exclamatory sentence expresses merely a feehng; only 
single words — Oh ! Alas ! etc., do that. The exclama- 
tory sentence expresses both a judgment and a feeling 
state of some sort about the subject-matter of that 
judgment. If I exclaim, "Oh, what a pain in my foot ! " 
" Oh, what a beautiful sunset ! " I mentally assert the ex- 
istence of pain, and the phenomenon of sunset; and I 
express at the same time certain feeling states which 
these objects excite in me. 

The interrogative sentence, it must be admitted, may 
also express a judgment. There are interrogative sen- 
tences which ask pure questions ; and a pure question 
is not a judgment ; but there are interrogative sentences 
which do not properly ask questions in the logical sig- 
nification of question. Such sentences express judg- 
ments as unambiguously as do declarative sentences; 
such are questions to which there is an expected answer, 
sentences in which information is not asked, but assent 
or denial is expected. The purpose of the questioner 
in such sentences is not to learn what the questioner 
does not already know, but to force some other person 
to accept a given judgment. 

The imperative sentence is the only one which does 
not express a judgment. This sentence can only ex- 
press a command. 



44 elementary logic 

Section ii 

the kinds of judgment 

There are three kinds of judgment : — 
(i) the Categorical; 

(2) the Hypothetical ; and 

(3) the Disjunctive. 

1. The Categorical Judgment. — In the categorical 
judgment the subject-matter is conceived and asserted 
as simple fact, as actual. For example, in the judgment 
"That cloud is dark," the simple existence of the cloud 
possessing a certain feature is asserted. In the judg- 
ment "All these men are honest," there is implied the 
actual existence of the individual beings called " these 
men." 

2. The Hypothetical Judgment. — It is the distinc- 
tive feature of this judgment that it asserts, not an actual 
fact merely, but a connection between a supposed fact 
and something which follows from this supposed fact. 
The essence of the hypothetical judgment is, therefore, 
supposition, and the development of the consequence 
of that supposition; this judgment makes a supposi- 
tion and develops its consequences. It is this feature 
which distinguishes the hypothetical from the categori- 
cal judgment ; the difference is that between actual fact 
and supposed fact. The nature of the hypothetical 
judgment will be best understood from an examination 
of some concrete instances of its use. 

As a first case let the judgment be, "If it rain, I shall 
remain at home." Here are two things: — 



JUDGMENTS 45 

(i) a supposed fact, a supposed state of the physical 
world; and 

(2) a consequence following from the supposed state 
of the world. Neither its raining nor my staying 
at home is regarded as actual; it is the connection 
between these two things that is asserted to be fact. 

Again, take the judgment, "If he is asked a favor, he 
will grant it." I do not assert that he is asked a favor 
nor that the granting of that favor is to be a fact ; but I 
do assert that these two things are so connected, that 
if asking the favor becomes a fact, the granting of that 
favor will become a consequent fact. 

Every judgment, we have seen, asserts something 
about the real world ; it is the very essence of a judg- 
ment to deal with reality, to claim truth. Now, in 
these two cases what is the reahty that is asserted? 
Plainly, not the raining nor the being asked a favor, nor 
my remaining at home, nor his granting this favor ; but 
a certain connection between the supposed things and 
something else; this is the real assertion made. Now, 
let us examine the assertion in each case, and we shall 
see that it means the following judgments : — 

(i) The real world has such a constitution that if 
such an event as its raining in this particular region 
occurs, a certain other event, viz. my not going out, 
will exist. 

(2) The real world is so constituted that given such 
a fact as this man being asked a favor, a certain other 
fact, viz. his granting this favor, will result. 

The hypothetical judgment, therefore, just as the 



46 ELEMENTARY LOGIC 

categorical, deals with the real world ; every such judg- 
ment asserts that the real world has a certain character 
or structure, so that if you assert or suppose a certain 
thing, some other thing will necessarily be a fact in this 
world. This interpretation brings to view another 
feature of the hypothetical judgment, and that is, this 
judgment depends upon an imphed categorical judg- 
ment. Unlike the categorical, the hypothetical judg- 
ment cannot stand alone; it implies a categorical 
judgment ; and the full meaning of this kind of judg- 
ment can be expressed only by adding an explicit 
categorical judgment. For example, the judgment, 
"If a body is allowed to fall freely, it tends toward the 
center of the earth," presupposes the categorical judg- 
ment, viz. the physical world is so constituted that in it 
there exists this necessary connection between a freely 
falhng body and its direction towards the earth's 
center. 

If, now, we compare the hypothetical with the cate- 
gorical judgment, we note this further difference in 
meaning and use, — the hypothetical judgment is ab- 
stract ; the categorical, concrete. I mean by this, that 
in judging hypothetically, we do not consider individuals 
or particular things as existing, but only as examples of 
universal laws or truths. The hypothetical judgment 
is used in asserting universal truths or laws of nature 
considered apart from any particular facts which illus- 
trate or fulfill these laws; the categorical judgment, 
on the other hand, asserts what is true of these particu- 
lar cases regarded as existing facts. 



JUDGMENTS 47 

This difference between the two kinds of judgment 
in question is thus a difference between two ways of 
deahng with the real world, — deahng with this world 
abstractly and dealing with it concretely. I deal ab- 
stractly with my real world, when I treat the various 
particular things in it, not simply as facts in themselves, 
but as instances of universal laws. Thus, in the judg- 
ment, " If a body is heated it expands,"! do not consider 
any particular heated bodies such as A, B, C, but a 
universal feature of the constitution of bodies. I deal 
concretely with the real world when I assert the exist- 
ence in it of particular things or classes of things, pro- 
vided I consider these classes as made of individual 
things which exist. 

It follows from this difference in the nature and func- 
tion of these two kinds of judgment that the hypothetical 
judgment always asserts what is universal. The cate- 
gorical, on the other hand, while it can assert what is 
universal, asserts also what is particular, which the hy- 
pothetical never does. 

This fact, that the categorical judgment asserts 
sometimes that which is universal, creates a difficulty 
in determining in some cases which of these judgments 
is made; for instance, is the judgment, "All men 
are mortal," categorical or hypothetical? In form it 
is categorical, but is it so in meaning? Does this 
judgment assert the mere fact that every individual 
man who has lived in the past, and every individual 
man now living, or who will ever hve, will die ; or does 
it assert a necessary connection between the nature or 



48 ELEMENTARY LOGIC 

attributes of man and the condition called mortality ; 
so that, given an individual having the attributes of 
man, you are certain he has also the attribute mortality ? 
I think there can be but little doubt that the latter is the 
meaning of this sentence. This judgment means, if an 
individual is a man, he will die. But take the judgment, 
"Every one present was delighted; " this judgment is 
both categorical and universal : it is categorical, because 
it distinctly implies the existence of the individuals 
themselves; it is universal, because it deals with a 
whole class of these individuals. 

But now compare this universal with the universal 
in the preceding judgment, and we remark this differ- 
ence in the universals: in the first case, the universal 
is made up of individuals who can be considered sepa- 
rately, and who can be counted or enumerated ; in the 
second case, the individuals are not so considered. 
Only one feature or characteristic of these separate 
individuals is considered, viz. that by virtue of which 
they are each instances of a universal law. The exist- 
ence of no one of these individuals is asserted or im- 
plied ; it is the reahty of a law only that is asserted. 

We accordingly distinguish two universals, — the 
enumerative universal, and the universal of law or 
necessary connection of attributes. The enumerative 
universal is concrete ; the universal of law is abstract. 

This difference of universals enables us to determine 
in any case whether the universal judgment is cate- 
gorical or hypothetical. If the universal is enumera- 
tive, the judgment is categorical ; if, on the other hand, 



JUDGMENTS 49 

it is the universal of law that is meant, the given judg- 
ment is hypothetical, albeit in form it may be the same 
as the categorical; thus, the judgment above, "All 
men are mortal," is a hypothetical judgment, because 
the universal here is that of law. The judgment asserts 
a necessary connection between the attributes connoted 
by the name man, and those connoted by the name 
mortal. Expressed in the form of a supposition, this 
judgment is, " If any being is a man, that being is mor- 
tal." On the other hand, the judgment, "All those 
present were delighted," is categorical, because it is 
an enumerative universal that is meant in this case. 

3. The Disjunctive Judgment. — This judgment as- 
serts that of two or more alternative possible things 
one is real. In this form of judgment, there are pre- 
sented to our thought alternatives; and alternatives, 
if taken seriously, exclude each other, so that if one is 
taken to be real or true, the others are not real at the 
same time. 

It must be noted that the disjunctive judgment does 
not assert which one of the suggested alternatives is 
real. It only asserts that some one of them is real. 
Thus, in the judgment, " A is either B, C, or D," the 
assertion is, that one of these alternative predicates 
goes with A ; but it is not determined which one it is. 
It is imphed, however, that if the real predicate is C, 
neither B nor D is the predicate of A at the same 
time. The disjunctive judgment, therefore, implies 
both knowledge and ignorance on the part of the one 
who judges. 



50 ELEMENTARY LOGIC 

This coexistence of knowledge and ignorance char- 
acterizes the situation in which one judges disjunc- 
tively. To judge, for instance, "A is either a knave or 
a fool, or a mixture of both," implies some knowledge 
of A's character; it implies knowledge enough to de- 
termine the possible predicates that can describe that 
character, — knavery, folly, or something of both ; 
at the same time, one who thus judges about A's char- 
acter confesses to some degree of ignorance, — the judge 
does not know which of these quaUties constitutes A's 
character. 

Two features, therefore, distinguish the disjunctive 
judgment from the other forms of judgment: — 

(i) This judgment deals with alternatives, with 
possibihties which are so related to each other that if 
one of them is asserted to be real, the others are ex- 
cluded from being real at the same time. 

(2) There is imphed in this judgment both knowl- 
edge and ignorance in the mind of the one who 
judges. 

To the exposition I have given it may be objected, 
that there are disjunctive judgments which, instead of 
implying partial knowledge and partial ignorance, 
imply complete knowledge of the subject-matter, a 
knowledge so complete as to exhaust all the possibili- 
ties; such, it may be said, are the judgments of classi- 
fication. For instance, when the geometrician says, 
"Triangles are either right-angled, obtuse-angled, or 
acute-angled," there is no ignorance whatever touching 
the subject-matter, but complete knowledge rather; 



JUDGMENTS 5 1 

SO that the classification is exhaustive. Two replies can 
be made to this objection : — 

(i) Assuming that this is a case of genuine disjunc- 
tive judgment, one which presents real options or 
alternatives, there is something undetermined, some 
element of uncertainty, some degree of ignorance im- 
plied in this case; the geometrician does not know 
what sort of a triangle he will next meet with, and he 
does not know into which of the classes he will put the 
next one to which his attention may be drawn. 

(2) But a better way of disposing of this difficulty 
is to point out the fact that this judgment is not really 
disjunctive in character; it is only a form of the cate- 
gorical judgment, the pecuHarity of which is, that in 
the form of a disjunction it merely divides the given 
subject-matter. There is no true disjunction in such 
cases; because no alternative possibiUties are pre- 
sented. This particular judgment, therefore, means 
that triangles are divisible into three classes; and that 
any given triangle belongs to one of these classes. 

Section 12 

the quality of judgments 

In respect to their quahty judgments are Affirmative 
or Negative. The distinction is that between affirming 
and denying, or negating. But, as we have seen, it is 
of the very essence of a judgment to affirm or assert ; 
the conviction of truth, the claim of reality, is the nerve 
of every judgment. How then can there be such a thing 



52 ELEMENTARY LOGIC 

as negation, the denial of reality in a judgment ? No 
conviction can be negative; it is present or it is not 
present; but if present at all, its nature is to claim 
reality, and this claim must be a positive thing. Never- 
theless, negation is a fact in our thinking ; and there are 
such things as negative judgments. Our problem then 
is to determine the nature of such judgments, the mean- 
ing of negation, and its function in our thinking. 

The first thing we observe, if we examine a negative 
judgment, is, that this judgment is not independent; 
it cannot stand alone, it is possible only on the basis of 
some affirmative judgment; he who denies can do so 
only as he first impHcitly affirms something ; the spirit 
that merely denies may exist in the world of fiction, but 
not in the world of logical thought, or in the world of fact. 
He who says, " There are no living beings in the moon," if 
he claims that this judgment is true, must affirm that the 
real world as known by him excludes living beings from 
that body ; he must be able to make positive assertions 
about the physical conditions of the moon in order to 
deny that there are living beings there. Every nega- 
tive judgment therefore implies, and rests for support 
upon, some affirmative judgment. But more than this ; 
we observe as the second fact, that negation has a 
positive use, and subserves a positive end in our 
thinking. 

Our thinking is always seeking truth and knowledge ; 
and the negative judgment is a stage in this progress 
of the mind toward knowledge. The special function of 
negation is to limit and define the direction in which our 



JUDGMENTS 53 

thought is going, or must go, if it is to reach its goal. 
Let me illustrate this function of negation, this service 
of the spirit that denies. I will suppose a traveler is 
in search of a certain town, which he knows is upon 
one of a number of roads, all but one of which roads 
diverge from the road on which he is proceeding; I 
will further suppose that, at each point where a road 
diverges, a guideboard informs our traveler that the 
town he is seeking does not he on that road. It is 
easy to see that these negative guideboards in the end 
give our traveler the information he desires ; they keep 
him on the right road ; they lead him to his goal ; they 
define for the traveler the object of his search; they 
do so by successive eliminations of all the roads that 
lead elsewhere. In other words, the object of the 
traveler's quest is progressively deiined to him; each 
of these eliminations of the wrong roads is a step toward 
the knowledge sought. 

Now, this is just the function of the negative judg- 
ment. We set out in our thinking in search of a cer- 
tain truth, a fact of some sort, just as our traveler 
seeks a certain place. This truth hes, at first, in dif- 
ferent possible directions; by negation we exclude one 
after another these suggested possible directions ; every 
excluded or negated one hmits the number of remain- 
ing directions, until the last negation itself brings us to 
our goal, and gives us positive possession of the truth 
sought. Thus is negation a method of determination 
or definition, the pecuharity of this method being that 
it proceeds by elimination. I can define such a 



54 ELEMENTARY LOGIC 

subject-matter as B in one of two ways : I can either 
give the positive marks which define and distinguish B ; 
or I can, by negative statements, ehminate one by one 
those which are not marks of B, and reach the same 
result, the definition or determination of B. 

Section 13 

other distinctions in judgment 

I. Analytical and Synthetic Judgments. — A judg- 
ment which analyzes a concept, or defines a name, is 
called by some logicians an analytical judgment, in 
distinction from judgments which unite to a subject 
something that is not already contained in the meaning 
of that subject. For example, the definition of a triangle, 
as a figure having three sides which inclose three angles, 
is, according to these logicians, an analytical judgment, 
because it only defines the meaning of a name, or 
analyzes the concept triangle ; while the judgment, " The 
sum of the angles of a triangle is equal to two right 
angles," is synthetic, because in this judgment something 
not already contained in the meaning of triangle is 
asserted of triangles as its property. In other words, 
in this judgment two distinct things are united, or 
synthesized, instead of one thing being analyzed. 
Doubtless there is a distinction between the two thought 
operations in these two judgments ; but this distinction 
really exists within every judgment ; in every judgment 
there are both analysis and synthesis; these processes 
are not separable in our thinking ; there is never analy- 



JUDGMENTS 55 

sis without synthesis, nor synthesis where there is no 
analysis. 

It is, however, true that in a given judgment, one or 
the other of these processes may be preponderant, and 
so give to this judgment its distinctive feature. The 
primary purpose of a thinker may be to analyze given 
subject-matter; and when such is the case, there is a 
propriety in calhng the judgment analytical; or, the 
main purpose of the thinker may be to determine the 
relation between things, and to proceed from one thing 
to another; in which case the judgment is fittingly 
called synthetic. 

2. Modal Judgments : Judgments of Fact, Judg- 
ments of Necessity, and Judgments of Possibility. — 
These distinctions are made by logicians; and they are 
generally assumed to correspond to differences in the 
mode of the assertion. Assertorial, Apodictic, and Con- 
tingent are also terms that designate these distinctions 
in judgments. 

The assertorial judgment makes a simple assertion of 
actuahty; for example, "Stars shine," is an assertorial 
judgment. The apodictic judgment asserts something 
to be necessary, or necessarily true; thus, "Things 
equal to the same thing must be equal to each other." 
The contingent judgment asserts what is possible ; for 
instance, "There may be living beings on the planet 
Mars." 

Let us examine these distinctions. And first, that 
between the assertorial and the apodictic judgment. 
Here are two judgments: "All the radii of a circle are 



56 ELEMENTARY LOGIC 

equal. " "All the radii are necessarily or must be equal. ' ' 
Now, wherein lies the difference between these two 
judgments? They both have to do with the same 
subject-matter. The conviction of truth expressed in 
the second is not more complete or stronger than the 
conviction expressed in the first. 

When the geometrician says, "All radii of a circle 
must be equal," he is not surer of that fact than he 
is when he merely says, "All radii of a circle are 
equal ; " he does not mean to add anything to the 
strength of his conviction by saying, "All radii must be 
equal." Yet there is a difference in the two judg- 
ments, or rather in the mental situations which are 
reflected in them. This difference is not strictly in 
the judgments themselves, as we have seen; it lies in 
something which is implied and not expressed ; and this 
implied thing is connected with the second judgment, 
the apodictic one ; this judgment contains a reference to 
something beyond itself; this is the force of the word 
must. When I assert, " All radii must be equal," I imply 
some kind of reason or ground on which this assertion rests. 

Further on we shall see that such judgments im- 
ply inference ; here, it is sufficient to observe that the 
distinctive feature of the apodictic judgment is, that it 
impKes a reason or ground on which its truth or validity 
rests; when we assert that something is necessarily a fact 
we imply that this fact is a consequence of, or is sup- 
ported by, some other fact. 

Let us examine next the so-called contingent judg- 
ment. We shall first observe that the words may and 



JUDGMENTS 57 

can which are signs of this judgment, are ambiguous. 
They are used to denote a state of uncertainty in the 
mind of the thinker, and they are used to express 
the possibihty of that which is the subject-matter of the 
judgment. Thus, when I say of a man who has been 
generally unsuccessful, ''He may succeed, and he may 
fail again," I mean to express my uncertainty as to the 
issue of his next venture. In this situation, I do not 
judge respecting this man's business ventures ; my judg- 
ment relates to my state of mind, and by impHcation 
I assert that I am not certain which of the two possible 
results, success or failure, is to be fact. 

Now take the judgment, "The planet Mars may be 
or can be inhabited by beings like ourselves ; " this 
judgment asserts a possible fact, not a doubting state 
of mind in the person who judges. 

We should also observe another meaning of may in 
propositions. For instance, a teacher says to his class, 
"The class may omit chapter four in preparation 
of the next lesson;" this judgment asserts neither 
uncertainty nor possibihty; it asserts rather permis- 
sion; its meaning is, the omission of chapter four 
is a permitted fact. The teacher really asserts a state 
of his own mind, a state of wilhngness in reference 
to a certain possible action of those students. 

Here, then, are three species of judgment, indicated by 
the words may and can; a judgment which asserts men- 
tal uncertainty, a judgment which asserts possibility, and 
a judgment which asserts consent or willingness. It is 
the second of these that is the contingent judgment of 



58 ELEMENTARY LOGIC 

the logicians, the judgment of possibiUty. Accordingly, 
let us next examine this judgment. 

Every judgment, we have seen, deals with the real 
world; every judgment asserts fact. How, then, can 
there be a judgment of mere possibihty, a contingent 
judgment? We saw in the case of the hypothetical 
judgment, that it presupposes, as its basis, a cate- 
gorical judgment. One can suppose something, and 
draw therefrom a consequence only in a real world. 
It is just so in the case of a contingent judgment: 
this judgment presupposes a categorical one; it as- 
sumes a real world of a definite constitution. Pos- 
sible things are thinkable, and can be asserted only 
in a world that is actual. When, therefore, the 
judgment is made, "Rational beings may inhabit the 
planet Mars," there is implied the judgment, "The 
physical universe has the sort of constitution which 
permits rational beings on the planet Mars." 

This, then, is the meaning, the distinctive trait of the 
judgment of possibility ; like the hypothetical and the 
apodictic judgments, it implies and is dependent upon 
another judgment. Contingency, possibility, are things 
which we can rationally think only as we think some- 
thing as actual, or real. 



CHAPTER V 
THE LOGIC OF PROPOSITIONS 

Section 14 
the meaning of the proposition 

In formal logic, a proposition is a sentence which 
expresses a judgment. A proposition consists of two 
parts, technically called the two terms, or names; a 
third constituent, the copula, is recognized by most 
logicians. 

A better analysis of the proposition is, to recognize 
in it two terms and a relation between them, these 
making the three elements of the proposition. The 
student must not identify the two terms of a logical 
proposition with the subject and predicate of the gram- 
matical sentence, nor suppose that the grammatical 
copula is identical with the relation between these two 
logical terms. The various parts of speech which 
grammar teaches us to distinguish in a sentence have 
no existence in the logical proposition. Either term in 
the proposition may consist of a single word, or any 
combination of words; and any word, no matter what 
part of speech it may be, noun, adjective, preposition, 
etc., can form the subject or predicate term of a propo- 
sition. 

59 



60 ELEMENTARY LOGIC 

These two terms in the proposition express the two 
concepts which are united in the judgments. And the 
assertion of the proposition is, that a definite relation 
exists between these terms. These relations are of all 
sorts, — space, time, quaHty, cause, hkeness, difference, 
etc. ; but, as we shall have occasion to explain later, 
for certain purposes in the use of propositions, these 
relations can be reduced to one or two, either the rela- 
tion of subject and attribute, or of class to class. Thus 
the proposition, " Heat expands bodies," is equivalent to 
either of these two propositions : " Heat possesses the 
property of expanding bodies," or " Heat belongs to the 
class of things which is characterized by the property 
of expanding bodies," that is, heat is one of the things 
which expand bodies. 

Again, logical analysis must not be identified with 
grammatical analysis. The logical analysis of a propo- 
sition consists simply in distinguishing the two terms 
(for convenience called subject and predicate terms), 
and the kind of relation which is asserted to exist 
between these terms. The logical analysis of a sen- 
tence which is not already in the form of a proposi- 
tion consists in reducing the sentence to the form of 
one or more simple propositions. In order to analyze 
a sentence logically, sometimes various changes are 
made in its wording, clauses being reduced to simple 
combinations of words, phrases to single words, verbs 
changed to other words or omitted altogether, as 
we shall show in the practical exercises under this 
topic. 



the logic of propositions 6l 

Section 15 
kinds of propositions 

Corresponding to the three essential kinds of judg- 
ments, logicians distinguish three kinds of propositions: 
the Categorical, the Hypothetical, and the Disjunctive 
proposition. Since a sentence which expresses a judg- 
ment is, from the point of view of logic, a proposition, 
whatever sentence expresses a categorical judgment, 
is a categorical proposition, whether this sentence be 
declarative, exclamatory, or interrogative. For the 
same reason both the conditional sentence of the gram- 
marians and the declarative sentence can be hypothetical 
propositions. The disjunctive sentence, the sentence 
with either, or, and their equivalents, is always a dis- 
junctive proposition. 

Two other propositions should be distinguished : prop- 
ositions which contain words implying exclusion, and 
propositions with exceptive or limitation words. These 
propositions are called Exclusive and Exceptive Propo- 
sitions. Some of their peculiarities deserve to be noted. 
The exclusive proposition is interpreted in two ways: 
either as equivalent to a single hypothetical proposition, 
negative in force, or as equivalent to two categorical 
propositions, the one affirmative and the other negative. 

Take the proposition, "Only members vote ; " this 
proposition is interpreted so as to mean that if a person 
is not a member, that person cannot vote; so inter- 
preted, the proposition says nothing about members; 
it does not assert that any member votes, nor that 



62 ELEMENTARY LOGIC 

no member or that some member votes; nor does 
it necessarily imply that at least some members vote; 
the entire assertion is about those who are not mem- 
bers ; and this assertion is that they are excluded from 
the class of voters. According to the other interpreta- 
tion, this proposition is equivalent to the two following : 
(i) " Those who are not members do not vote." (2) 
" Some members at least do vote ; " and according to this 
view, the proposition asserts that non- members are non- 
voters ; and implies that some members are voters. 

The first of these interpretations hardly seems ad- 
missible. A situation is hardly conceivable in which 
it would be said that only members vote, if in fact 
no members were voters, or had not the right to vote. 
The proposition limits the right of voting to mem- 
bers ; such is the force of the term only. It would 
be meaningless to announce that only holders of 
certain tickets could occupy seats in a grand stand, 
if it were not the intention of the managers to admit 
to those seats any who did hold these tickets. The 
exceptive proposition is equivalent to two propositions ; 
thus, the proposition, "All but five were drowned," 
means five were not drowned; all the rest of that 
company were drowned. 

Section 16 

the quality of propositions 

Negative Propositions. — We have seen what negation 
is in judgment. It now remains to consider in what 



THE LOGIC OF PROPOSITIONS 63 

way negation is expressed in propositions, and con- 
sequently what propositions are negative in their 
quality. A Negative proposition is one which asserts 
either an absolute difference between two things, or an 
exception to a rule or general statement ; for example, 
"Birds are not mammals" is a negative proposition, 
because it asserts an absolute difiference between these 
two classes of living beings. " Right is not might" asserts 
a complete difference between these two things. " Some 
mistakes are not culpable" asserts an exception to the 
rule or the general statement, that mistakes are culpable 
things. Some one asserts, "Every man has his price; " 
the reply is, ''One man, Mr. A. has not his price;" in 
other words, Mr. A. is an exception. 

The student must be admonished that not all nega- 
tive words in a proposition are signs of negation ; also 
that some propositions which contain no negative words 
are either negative themselves or imply negative propo- 
sitions. As an instance of an affirmative proposition 
in which there are negative terms, take the following, 
"What is not an animal is not a man;" in this propo- 
sition both the subject term and the predicate term are 
negative; and yet the proposition is not negative, since 
there is no real negation in the judgment. 

A proposition is negative only when the negation 
affects the assertion. In the above proposition it 
affects only the terms; the proposition "Animals are 
not men," is a negative proposition ; for here the nega- 
tion is in the assertion. As instances of propositions 
whose terms are positive but whose force is negative 



64 ELEMENTARY LOGIC 

take the following: "Few shall part where many meet," 
"Few are acquainted with themselves," " Perchance 
for a good man some would even dare to die. " These 
propositions are negative in force, although they con- 
tain no negative terms. I think every one would ad- 
mit that they contain negation rather than affirmation ; 
" the few that part " are an exception to the rule. 
The force of the judgment is most do not part in the 
situation described by the poet. The clear implica- 
tion of the proposition, "Few are acquainted with them- 
selves," is, that most are not acquainted with themselves. 
That is what the writer means to say in an emphatic 
manner. 

In the last example there can be no question about 
the quaUty of the proposition. Paul meant to assert 
that men as a rule are not wilUng to die, even for a 
good man. It should be noted that the hypothetical 
proposition is negative only when if is the connection 
between the protasis and the apodosis that is denied. 
For instance, the proposition, "If he did not commit this 
crime, he is safe," is affirmative; since the negation 
affects only the protasis ; on the other hand, the propo- 
sition, "If he did commit this crime, he will not escape," 
is negative; because it is the connection between the 
protasis and the apodosis that is negated. 

Finally, it must be borne in mind that the disjunctive 
proposition can never be negative, since in that case 
disjunction or alternatives would be impossible. To 
negate in the disjunctive proposition is to destroy the 
proposition itself. The proposition, "A is neither B 



THE LOGIC OF PROPOSITIONS 6$ 

nor C," is a negative proposition, but it is not a dis- 
junctive proposition, for the reason that it really pre- 
sents no alternative. 

Section 17 
the quantity of propositions, quantification 

The Quantity of a proposition means the extent or 
scope of the assertion ; and, since two terms constitute 
a proposition, quantity appUes to both terms, and not 
to the subject term only, as formal logicians for the most 
part maintain. In its application to the subject term, 
quantity signifies the extent or amount of that which is 
considered in the assertion, or of that to which the pred- 
ication applies. 

In its application to the predicate term, quantity 
means the extent to which the predicate term is appUed 
or used in the predication ; thus, in the proposition, 
"Fixed stars are self-luminous," the predicate term 
self-luminous is not appUed in its full extent to this 
subject; because, while the quaUty of being self-lumi- 
nous belongs to all fixed stars, it may also be a quality of 
other bodies; but in the proposition, "Only the brave 
deserve the fair," the predicate term, deserve the fair, 
is appHed in its whole extent, because this quaHty is 
Umited to the brave. 

In formal logic, propositions are distinguished as 
Universal and Particular. This distinction is based 
upon the quantity of the subject term only. A uni- 
versal proposition is, accordingly, one in which the 



66 ELEMENTARY LOGIC 

predication applies to the entire subject term. A 
particular proposition is one in which the predication 
is appUed to less than the entire subject term. Ac- 
cording to this distinction the singular and the col- 
lective propositions are universal in their quantity, 
because the predication is apphed to all that is named 
by their subject terms; for example, "The North 
Pole has not yet been reached," is a universal propo- 
sition. 

The signs of universality in propositions are such 
words as all, each, every, the whole, etc. The signs 
of particularity are such words as some, few, mosi, 
many, a part of, etc. These signs must not be con- 
founded with other words which imply generality 
or particularity, such as always, ever, never, some- 
times, rarely, etc. These words do not affect the 
quantity of the proposition; for example, the propo- 
sition, "All men sometimes do wrong," is universal; 
while the proposition "Some men are always in the 
wrong," is a particular proposition; since the words 
sometimes and always affect the assertion, and not the 
quantity of the subject and predicate terms. The first 
proposition asserts that doing wrong sometimes is a 
quality or mark of all men ; the second proposition as- 
serts that being in the wrong at all times is the quality 
or mark of some men. 

There is another error against which the student 
must be admonished: the confusion of the quantity 
of the subject term with that of a larger class or whole 
of which this subject term even taken in its full ex- 



THE LOGIC OF PROPOSITIONS 6/ 

tension is a part, and which the subject term implies 
or suggests. For instance, "All male citizens have 
the right of voting," is a universal proposition; al- 
though its subject term clearly implies a larger class in 
which the class, all male citizens, is included, but in the 
proposition it is the male citizens only which are con- 
sidered and to which the predicate, have the right to 
vote, can be given; none of the other inhabitants are 
considered in the proposition. 

Quantification of Propositions. — So far I have set 
forth the customary doctrine which limits quantifica- 
tion to the subject term, and determines the quantity 
of the proposition by the quantity of this term only. 
But quantity is not a feature of the subject term only ; 
it belongs to the predicate term as well; and, since 
in a proposition both terms must be considered, it 
is impossible to disregard this quantitative aspect 
or significance of the predicate term, when we are 
determining the exact scope or range of the assertion. 
For this reason it is no less important to consider 
the extension of the predicate term than to consider 
the extension of the subject term. In determining the 
exact scope of the proposition, "All men are mortal," 
it is not unimportant to consider whether this attri- 
bute of mortality is limited to men, or whether it may 
be also a quality of other beings. 

By the quantity of a proposition, therefore, I shall 
mean that scope or extent of the assertion which is 
determined by the extension of its subject and predi- 
cate terms. Quantification is a method of treating a 



68 ELEMENTARY LOGIC 

given proposition so as to make explicit its quantity 
in the sense defined above. This method is artificial, 
and to subject propositions to it, is in some cases to 
create quite unnatural and rather awkward forms of 
statement; but this method is justified by the result 
attained, which is the exact determination of the mean- 
ing of the proposition. 

Quantification is most easily effected if we conceive 
the relation between subject term and predicate term 
to be a relation between two classes; this relation in 
any given proposition will always be one of inclusion 
of the subject class within, or exclusion from the 
predicate class; and this inclusion or exclusion will be 
of the whole class, or of a part of it only. Making, then, 
the class relation the basis of quantification, we can 
easily see that the different relations as respects quan- 
tity which can exist between the subject class and the 
predicate class are the following : — 

(i) All of the subject class may be included in 
the predicate class, so as to exhaust that class, leav- 
ing no room in it for any other class. 

(2) All of the subject class may be included in the 
predicate class, yet so as to leave room for other 
classes, the subject class forming only a part of the 
predicate class. 

(3) Some part of the subject class may constitute 
the whole of the class named by the predicate term. 

(4) Some part of the subject class may be in- 
cluded in the predicate class, yet so as to leave 
room for other classes. 



THE LOGIC OF PROPOSITIONS 69 

(5) Only some part of the subject class may be 
included in some part of the predicate class. 

(6) All of the subject class may be excluded from 
the predicate class. 

(7) Some part at least may be excluded from the 
predicate class. 

(8) Only some part of the subject class may be 
excluded from the predicate class. 

A convenient device for exhibiting these different 
possible relations between the subject and predicate 
classes is to use circles to represent the two classes, 
and the letters S and P to distinguish the subject and 
predicate classes. The different arrangements of these 
circles, together with the different positions of the 
letters S and P, will symbolize the quantification of 
any proposition ; for example, the following diagram 
exhibits the quantification of the proposition, "All 
men are some part of the class called mortals." 

The arrangement of the two circles shows that while 
all of the circle 5 is within the P circle, it does not ex- 
haust that circle; there is room for 
other circles within it. P is placed in 
two positions to show that while the 
circle 5 is coextensive with some part 
of the P circle, it leaves some part 
of that same circle unoccupied. Take 
as another illustration of this use of circles, the 
proposition, "All of A's times of being present in a 
certain place are identical with all of B's times of being 
present in that place;" to exhibit the quantification 




70 



ELEMENTARY LOGIC 



of this proposition the circles and letters are placed 
in this way : — 

The two circles are concentric, and of equal extent. 
Let us now use the letters 5 and P for the subject and 

O predicate classes in the eight statements 
made above, and we shall get the follow- 
ing formulae for expressing quantifica- 
tion : — 

1. All of S is all of P. 

2. All of .S is some part of P. 

3. Some of S is all of P. 

4. Some of S is some part of P. 

5. Only some part of S is some part of P. 

6. No part of 6' is any part of P. 

7. Some part of 5' is not any part of P. 

8. Only some part of 5 is not any part of P. 

The student will see that the following arrangements 
of the circles and the letters 5 and P symbolize these 
formulae : — 




Fig. I. 



Fig. 2. 



Fig. 3. 




Fig. 5. 



Fig. 4. 




Fig. 6. 



THE LOGIC OF PROPOSITIONS 7 1 





Fig. 7. Fig. 8. 

Inspection of these diagrams shows the following things 
which the student should keep in mind in constructing 
similar ones : — 

(i) There are but four different arrangements of the 
circles themselves, as in figures i, 2, 4, and 6. 

(2) In formulae 2 and 3 the positions of the letters 5 
and P are reversed. 

(3) In formulae 5 and 8, portions of the circles are 
shaded, to indicate that it is only those parts of the circle 
to which the statements apply. 

To make clear this matter of quantification, and to 
show the application of these formulae, I will now quan- 
tify a few propositions. 

(i) "All men love happiness." In this proposition 
the quantity of the subject term is already explicit, 
but not so the quantity of the predicate term; this 
does not show by its form whether all men make up 
the entire class of those beings who love happiness, 
or only a part of that class. Accordingly, I change 
the proposition in form so that it reads, "All men are at 
least a part of the class of beings that love happiness. " 
In other words, I apply to this proposition formula 2, 
substituting the subject and predicate terms of this 
proposition for S and P of the formula. 



72 ELEMENTARY LOGIC 

(2) "A and B are always together." This proposi- 
tion asserts that all of ^'s times of being present are 
coincident with all of 5's times of being present. Sub- 
stituting the terms of this proposition for the iS and P 
in formula i, we get "All of A is all of 5." 

(3) "Most men are honest." Applying to this 
proposition formula 4, it becomes, "Some men are 
some part of the class called honest beings." 

(4) "Only some of those present took part in the 
sports." This proposition comes under formula 5, 
"Only some part of S is some part of P." The par- 
ticipation in the sports was limited to a part of those 
present ; hence, formula 5 applies to this proposition. 

(5) "Only members vote." If we make this propo- 
sition equivalent to two propositions, one of them will 
be, "Some members at least are voters," and the other 
will be, "No other persons are voters." The first of 
these propositions is quantified according to formula 4, 
and it becomes, "Some of the members are some part 
of the voters"; the second of these propositions comes 
under formula 6, and it becomes, "No non-members 
are any part of the class of voters." Instead of resolv- 
ing this proposition into two others, we can quantify it 
under formula 3, and it will read, "Some, at least, of 
the members are all the class of voters." 

(6) "It was only some of the candidates who did 
not pass in the examination." Formula 8 apphes to 
this proposition; and the resulting proposition is, 
"Only some of the candidates are not any of those who 
passed in the examination." 



THE LOGIC OF PROPOSITIONS 73 

These examples should sufficiently explain the method 
of quantifying propositions. The importance of this 
treatment of propositions can only be appreciated when 
we come to study the relation between propositions and 
the processes whereby we pass from one proposition to 
other propositions, as we do in reasoning. And it is to 
the study of these processes that we now proceed. 



CHAPTER VI 

INFERENCE, REASONING 

Section i8 
the nature of inference 

Inference is the act of proceeding from one or more 
given judgments to some other judgment. It is the 
acceptance of a judgment because of its connection 
with some other judgment already made. To infer is 
to beheve something on account of its connection with 
some other thing. This something can be either a fact 
of experience, or an abstract idea or judgment. 

I. The essential elements of inference are three: 

(i) The datum or premise; 

(2) The conclusion or result reached; 

(3) The basis or ground of the inference. 

The datum or premise is that from which the infer- 
ence proceeds, its terminus a quo. The conclusion is the 
terminus ad quem of the process. The basis or ground 
is the reason or justification of the inferential process, or 
of the conclusion reached. The ground of inference 
must not be confounded with the datum or premise. 
Logicians have not always avoided this confusion ; but 

74 



INFERENCE, REASONING 75 

misconceptions of inference arise from the failure to 
distinguish these two things. The ground of the infer- 
ence is not always, indeed is rarely, expressed ; it is 
either implicit in a premise, as we shall see is the case in 
syllogistic reasoning, or this ground is a postulate or 
assumption that is tacitly made in proceeding from 
datum to conclusion, as in inductive inference. 

Some examples will best make clear what I mean by 
this ground of inference and its distinction from the 
other two constituents of the inferential process. 

Take as the iirst case the classic syllogism, "All men 
are mortal; Caius is a man; therefore, Caius is 
mortal." The first two propositions are the data or 
premises of this reasoning; the ground or justifica- 
tion of the inference that Caius is mortal hes in the two 
propositions which constitute the data. Examination 
of these two propositions discovers that this ground or 
reason for the conclusion that Caius is mortal, is the 
identity of the attributes of Caius and the attributes of 
all men, with which attributes mortality is connected. 
Caius is mortal because he has those human attributes 
which are connected with mortality. 

As a second example, take the following: "^ has 
died, B has died, C has died, etc. ; hence, I infer that 
X, being a man, will also die." The ground of this 
inference is the postulate or assumption of uniformity 
of experience. This assumption is, that these cases of 
mortahty of A,B, C, etc., are instances of some univer- 
sal law ; and, therefore, if X is like A , B, and C, he will 
die also. 



76 ELEMENTARY LOGIC 

2. The Criteria or Tests of Inference. — Inference 
must first be distinguished from other mental opera- 
tions that on a superficial view seem to be the same. In- 
ference is sometimes identified with association of ideas. 
This is a mistake ; the mental operations are different. 
The characteristic of thinking is the discernment of 
relations between things. In thinking about things, we 
do not merely take note of them as events that occur; 
we do not merely perceive them as present facts, or re- 
call them as past facts, or picture them as future facts ; 
we perceive relations of various sorts between these 
things — relations of time, space, quantity, causation, 
etc. ; and it is these relations themselves and not merely 
related things that are the subject-matters of our 
thinking. Mental association can deal with its objects 
only as concrete things, and as wholes ; and it couples or 
connects its objects only in one way, that of conjunction, 
or succession in experience. Thinking deals with its 
objects differently; it analyzes them, abstracts quahties 
from things, relations from related things ; and it unites 
things by relations quite other than those of mere as- 
sociative connections. 

Familiar instances will make clear this differentia of 
inference. A cat opens a door by jumping up and 
moving the latch. A dog, accustomed to go with his 
master in a boat, is told to get a sponge, goes to the 
house, and returns with it. A very young child puts its 
finger in a flame, gets burned, and next time avoids the 
flame. These are all situations in which a logical 
thinker would or could reason. The cat and the dog and 



INFERENCE, REASONING jy 

the child deal with data, with things which could be 
premises for a reasoner. They interpret in some fash- 
ion these data, and they reach conclusions or results 
of a practical character; and these conclusions could 
be formulated in propositions. It is quite certain 
that neither the cat nor the dog nor the child has 
performed an act of logical inference in these situations. 
All that is necessary to credit the cat with doing is, at 
most, a recall by associative memory of other instances 
in which it opened the door by hitting the latch ; this 
hitting of the latch having been in the first instance 
purely accidental; the successful hits become asso- 
ciated with opening the door, and the cat acts upon this 
associative connection. The same explanation applies 
to the case of the dog. Accustomed to bringing things 
in response to words, gestures, etc., and accustomed 
to seeing water removed from the boat by the use of a 
sponge, the dog sees the water in the boat, does not see 
the sponge there, hears the sounds, sees the gestures 
which are associated in his experience with bringing 
objects, and brings the sponge. Here, as in the case 
of the cat, the mechanism of association, memory im- 
ages, and motor reactions associated with them solve 
the problem. Mental operations of the same sort ex- 
plain the action of the burned child that dreads the fire 
and avoids contact with it. 

In these cases, while the mental processes lead to 
the same practical consequences that would follow 
from inference, they are not inferences ; and for the fol- 
lowing reasons : — 



78 ELEMENTARY LOGIC 

(i) There is in these cases no perception of the 
essential property or relation on which the consequent 
action depends. The cat does not perceive the 
essential property of the latch on which opening the 
door depends, nor does the dog perceive the • prop- 
erty on which getting water out of the boat depends, 
nor does the child discern this property in the flame 
that burned its finger. Put a reasoner in these situa- 
tions, and he would devise a way of opening the door if 
the latch were out of order ; he would get a dipper, if 
the sponge could not be found; and he would avoid 
touching all hot objects as well as the flame. 

(2) There is no genuine inference in these cases, 
because there is no consciousness of the necessary 
connection between data and result, between prem- 
ises and conclusion. It is not seen that these conse- 
quences and no others will follow from the data or 
given premises. The cat does not mentally say, ''If I 
do this, that must be the result." The dog does not 
virtually say, "If I get something that will hold water, 
that water in the boat can certainly be removed." But 
this is just what the logical thinker does say, viz. if a 
certain thing is done or is a fact, some other thing or fact 
will be the necessary consequence. 

It is this discerned connection between premises 
and conclusion, and the acceptance of the conclusion 
solely because of this discerned relation it sustains to 
the premises, that constitutes an act of logical infer- 
ence. For logic, there is no unconscious reasoning; 
the function of logic is to make us conscious of our 



INFERENCE, REASONING 79 

thinking ; to think in the logical meaning of the term 
is to deal thus critically with our thinking, to be aware 
of what we are doing, and why we are doing it. One 
criterion, therefore, of genuine inference is that the con- 
clusion is accepted because of its connection with the 
premises ; and therefore we must be conscious of this 
connection. 

This first differentia of inference will be better ap- 
preciated if we distinguish between inference as logic 
deals with it and inference in its psychological char- 
acter or aspect. This difference is one of function 
and aim. From the point of view of psychology, infer- 
ence, like other mental processes, is simply a fact or 
phenomenon to be described, and to be explained as 
science explains all phenomena. From the point of 
view of logic, inference is judged and valued according 
to its fitness to attain a certain end. The psychologist 
describes inference ; the logician evaluates it. For the 
psychologist, it is a matter of indifference whether a 
given inference is correct or incorrect ; for the logician, 
the character of the inference is a matter of fundamental 
importance. It is the function of psychology to de- 
scribe the way in which inference takes place as mere 
mental process; it is the function of logic to ascertain 
how this process must go on, if it is to attain its end. 
Psychology is thus merely descriptive; logic is regula- 
tive or normative in its treatment of thinking. Psy- 
chology is not concerned with such matters as truth, 
vahdity, etc. ; for logic, these aims and ideals are a pri- 
mary concern. 



80 ELEMENTARY LOGIC 

Logical thinking marks an advanced stage in mental 
development; men reason, and reason correctly, long 
before they attain this stage of thinking. Most men 
who reason correctly do so without knowing why their 
reasoning is correct, just as people for the most part 
take food, and the right sort of food, without knowing 
how it nourishes them, or why one sort of food is good 
for them and another sort bad. The business of logic 
is to make one conscious in his thinking, to make one 
know when and why this thinking is correct. Here Hes 
the practical function of logic. On its negative side, this 
function is to safeguard us from error ; on its positive 
side, the function of logic is to guide us in the search 
for truth and knowledge. 

The second criterion or test of genuine inference is 
that it must give as a conclusion something which is so 
far distinct from the data or premises, that it would 
not be perceived, but for the mental processes involved 
in inference. The conclusion must in some sense be 
contained in, or justified by, the data; otherwise it 
cannot be gotten from those data. The question is, 
how different from the premises must the conclusion be, 
to make the processes of obtaining it one of inference? 
Some logicians, John Stuart Mill, F. H. Bradley, and 
others, so emphasize this novelty in the conclusion as to 
reject the inference of formal logic for the reason, they 
maintain, that the conclusions in these inferences are 
not a distinct advance upon the premises, are not new 
facts which add something to the knowledge already 
had. 



INFERENCE, REASONING 8 1 

Hence, these logicians maintain that the syllogism 
of traditional logic is not a genuine form of inference, but 
only the appearance of inference. They say the syllo- 
gism begs the question, because it gives nothing in the 
conclusion that was not already contained in the given 
premises ; the conclusion is only an explicit statement 
of what is impHcitly asserted in the premises; it is like 
taking from a drawer something which you have first 
put there, or from a memorandum book some fact you 
have previously recorded there. The drawer is a use- 
ful thing, the memorandum book a serviceable thing; 
but they do not add to your possessions or to your store 
of known facts. Inference, say these logicians, must 
bring to light a new fact, must yield a judgment that is 
not already contained in the judgments from which it 
is derived. 

Now these logicians are right in maintaining that there 
must be a real difference between conclusion and data. 
It is concerning the character and amount of this differ- 
ence that the controversy between Mill and the defend- 
ers of traditional logic is waged. Mr. Mill, I think, 
exaggerates the amount of this difference; he misap- 
prehends the nature and the principle of syllogistic 
reasoning, and he fails to discern the true character of 
the inference which he accepts as genuine. 

Let us examine again the process of inference. In 
the first place, it is clear that this difference between the 
premises and the conclusion cannot be absolute, or 
there would be no connection between them, hence, no 
inference at all. Inference impUes some continuity in 



82 ELEMENTARY LOGIC 

thinking, and continuity involves some element that is 
identical throughout the process, otherwise the succes- 
sive steps cannot be linked. Any case of inference, if 
examined, will show something that is identical in prem- 
ises and conclusion, or some element of identity in 
the connection between them; this is as true of the 
instances Mr. Mill gives — of genuine inference — as 
it is of those he rejects as spurious inferences. 

To take one of Mill's examples, "Peter, James, 
John, etc., are mortal; therefore Caius is mortal, 
or all men are mortal." Now, Mill does not main- 
tain that the mortality of Peter, James, John, etc., 
is the logical ground of our expectation that Caius 
will die, or of the conclusion, "All men are mortal." 
The logical justification of this belief is, even accord- 
ing to Mill, the assumed identity of essential attri- 
butes possessed by these men, those who have died, 
and those we beHeve will die. Mill does say that the 
only reason we have for beUeving that Caius will die, 
or that all men will die, are the particular facts of 
observation, the death of Peter, James, John, etc. But 
in this statement Mill has confounded datum with the 
ground of inference. It is not these particular facts 
as mere particulars that justify this inference to other 
facts, but the assumption that these particulars are 
instances of a universal law of necessary connection 
between the attributes of man and mortality. The 
reason, therefore, for the beUef that Caius will die is, 
that he possesses those attributes which are thus uni- 
versally connected with mortality. 



INFERENCE, REASONING 83 

If Mill had not confounded datum with ground 
of inference, he would not have made this erroneous 
assertion. Mill came near the truth in making the 
uniformity of nature the foundation of inductive in- 
ference; but he misapprehended the connection be- 
tween this principle and inference; and, as I shall 
point out later, he did not rightly understand the prin- 
ciple itself. 

Now, let us take an instance of the reasoning Mill 
rejects. "All men are mortal; Caius is a man, there- 
fore Caius is mortal." Mill maintains that we have 
here no genuine inference, because the conclusion only 
asserts in explicit form what is already contained in the 
proposition, "All men are mortal." Mill is quite right, 
if this proposition is the sort of universal that he assumes 
it to be, — the enumerative universal. If this propo- 
sition, therefore, merely asserts that every individual 
man who can conceivably be counted either has died or 
will die, such a universal does contain Caius; and the 
proposition in which it is the subject term does, as Mill 
contends, assert the mortality of Caius ; and the propo- 
sition which professes to be the conclusion only says the 
same thing a second time. But Mill failed to distin- 
guish between two universals which are used in the 
syllogism, and therefore he failed to distinguish between 
a genuine inference in the form of a syllogism and a 
spurious inference. 

Now, the universal proposition in this argument 
asserts a law of connection between the attributes of 
man and mortaHty ; and the next proposition asserts 



84 ELEMENTARY LOGIC 

that Caius possesses these human attributes; and the 
third proposition asserts a genuine conclusion, just 
as genuine a conclusion as the one Mill draws from 
the premises, "Peter is mortal, John is mortal." This 
syllogism, therefore, is not open to Mill's objection, 
that it goes in a circle or begs the question. There is, 
we admit, an unproved major premise in the reasoning ; 
so is there something assumed and not proved in the in- 
ference Mill accepts. I maintain, therefore, that this 
syllogism gives us a genuine inference, because the con- 
clusion is something which is not recognized and cannot 
be justified until there has been some thinking done 
upon the datum. I cannot say Caius is mortal until I 
see that he is a man. I first link man with mortality, 
then Caius with man, then Caius with mortality. 

Let us examine next some cases of inference accord- 
ing to formal logic against which Mill's criticism seems 
better justified. "All ^'s are B's ; therefore, no A is not 
B, or what is not B is not A, or some B's must be ^'s." 
Surely, with more justice can one say these are not cases 
of inference. Each one of the propositions, after the 
first, only asserts the same thing in a different form of 
statement; there is identity of meaning under differ- 
ences of expression. But is this identity of assertion 
apparent without some constructive effort, some think- 
ing that discovers the substantial identity in these 
propositions? Grant that these propositions are all 
implied in the original one, and that the processes by 
which they are derived from it are only ways of develop- 
ing the impHcations of this datum. Still, I think we 



INFERENCE, REASONING 85 

must admit that these processes add something to that 
datum, to what we saw in it or took it to be at the out- 
set. 

How many young students in logic would be able 
to recognize the identity of meaning in this group of 
propositions without considerable mental effort? Do 
not the subsequent propositions add something to the 
content and scope of the original proposition, so that 
they are not a mere repetition of the same proposition, 
with sUght differences in wording ? I think the answer 
must be in the affirmative, and that the formal logicians 
are in the right in regarding these operations as infer- 
ences. Of course, such inferences are of a different 
sort, and are based upon a dififerent principle from the 
inferences which Mill regards as the only genuine ones. 
And this fact naturally leads to our next topic. 

Section 19 

the forms of inference 

It is customary in logic to distinguish two kinds of 
inference, — Deductive and Inductive. These two 
kinds of inference differ first in form. In the de- 
ductive inference, at least one of the premises is a 
universal proposition, and the conclusion is a proposi- 
tion of less extension than the premises ; hence the name 
deductive, which imphes a leading down from a uni- 
versal or general truth to a special case. In inductive 
inference the premises are particular propositions, 
and the conclusion can be universal or particular. 



86 ELEMENTARY LOGIC 

Inductive inference proceeds either from particulars to 
a universal or to other particulars. 

The second difference between these two kinds of 
inference is more important ; it is a difference in func- 
tion and aim. The function of deductive inference 
is to estabhsh consistency between judgments; the 
function of inductive reasoning is to attain knowledge 
of facts. Consistency is the aim of deductive reason- 
ing, as it is employed in formal logic. Truth and 
knowledge are the aim of inductive reasoning. For 
deductive reasoning, it is indifferent whether the prem- 
ises are true or false propositions ; its only concern is 
to reach conclusions which are consistent with the given 
premises, — true propositions if the premises are true, 
false conclusions if the premises are not true. Induc- 
tion, on the other hand, starting with facts of experi- 
ence, aims to enlarge our knowledge by reaching other 
facts not observed. 

Connected with this difference in function and aim is 
a third difference between deductive and inductive in- 
ference, — difference in the subject-matters with which 
they deal. The subject-matters in which deductive 
reasoning is employed are abstract things for the most 
part. The subject-matters about which inductive rea- 
soning is employed are concrete things; objects of 
perception are its data, facts of actual or possible ex- 
perience are its conclusions. Hence, deductive infer- 
ence belongs to formal logic; inductive inference, to 
science. 

But there is a fourth and a more profound difference 



INFERENCE, REASONING 8/ 

between these two kinds of inference; it is a differ- 
ence which has to do with the inferential processes 
themselves, with the nature of the connection between 
premises and conclusions, and with that which I have 
called the ground of inference. The conclusion in 
inductive inference goes beyond the premises to an 
extent and in a way which is not the case with the 
conclusion in deductive inference; thus, when I con- 
clude from the fact that several substances exhibit a 
certain property, that all other substances which are 
hke these will exhibit this same property, this conclusion 
goes directly beyond these data of experience. It is a 
step from the known to the unknown, as Mr. Mill 
rightly says. On the other hand, when, from the two 
propositions, "A is B," "C is A," I conclude "C is 5," 
there is no such going beyond my premises; there is 
no step from the known to the unknown, as there is in 
the first case. 

Again, the connection between premises and conclu- 
sion is of a different sort in these two inferences. The 
proposition, "All other substances which are like these 
known substances will behave in the same way," is not 
connected with the premises from which it is drawn, in 
the same way as is the conclusion, "C is 5," with its 
premises. In other words, I beUeve the conclusion in 
the first inference for a different reason than the reason 
which compels me to accept the conclusion in the second 
inference. 

The reason why I believe that all substances not 
yet observed will behave as do these substances 



88 ELEMENTARY LOGIC 

which have been observed, is an assumption of some 
causal connection, some uniform way of acting on the 
part of nature. The reason why I am certain that C is 
By is that, analyzing and putting together the two given 
propositions, I discover that the properties which make 
C and A are identical with the properties which make 
every A a, B; it is this identity of attributes that links 
C and B ; and I can therefore say the proposition "C is 
5" is contained in or is impUcated in the other two 
propositions, and analysis discovers this fact. Accord- 
ingly, we can say deductive inference is inference by 
implication ; induction is inference based upon assumed 
causal connection. 

Finally, it should be noted that the conclusions in 
deductive inference are certain and necessary. In in- 
ductive inference, on the other hand, the conclusion is 
only probable. The explanation of this fact is found 
in the differences I have already explained. Deduc- 
tive reasoning, since it deals with our thoughts and their 
connections, and since it aims only to make our thoughts 
consistent, must reach conclusions that are certain. 
Inductive inference, on the other hand, having to do 
with matters of fact, with things which we can know 
only through experience, being a step into the unknown, 
cannot give conclusions that are certain. In the world 
of experience, in the world of actual things and events, 
probability, not certainty, is the best we can attain. 

Each of the two kinds of inference we have dis- 
cussed presents two varieties. Deductive inference is 
either Immediate or Mediate. Inductive inference sub- 



INFERENCE, REASONING 89 

divides into generalization of experience or inductive 
generalization, and inference by analogy. The dis- 
tinction between immediate and mediate inference is 
this: in immediate inference the conclusion is drawn 
from a single proposition; in mediate inference two 
propositions are necessary for the datum. Immediate 
inference is direct ; it consists of but one step. Mediate 
inference is indirect ; there are two steps in the process. 
The two varieties of inductive inference will be con- 
sidered in Part Two, since they belong to the Logic of 
Science. 



CHAPTER VII 

THE FORMS AND METHODS OF DEDUCTIVE 
INFERENCE 

Section 20 

equipollence 

Immediate inference is based upon two distinct rela- 
tions that exist between any two propositions, — Equi- 
pollence and Opposition. By the equipollence of two 
propositions is meant that they express essentially the 
same judgment, but under such different forms of state- 
ment as to require some rejflection to recognize this 
equivalence of the two propositions. Thus, these two 
propositions are equipollent, "All men love happiness," 
"Some beings who love happiness are all men." One 
and the same judgment is expressed in these two sen- 
tences, but this identity is not at first perceived; its 
recognition involves some exercise of thought; it is a 
something that is reached by a process which starts with 
a datum, and we have seen that such a process is an 
inference. Hence, for example, from the proposition, 
"AH men love happiness," we infer the truth of this 
equipollent proposition, " Some beings that love happi- 
ness are all men." 

90 



DEDUCTIVE INFERENCE 9I 

Equipollence is maintained in four ways : — 
(i) By obversion, 

(2) By conversion, 

(3) By contrapositions, 

(4) By added determinants. 

I will explain these methods of immediate inference 
in the order enumerated. 

I. Inference by Obversion. — This means a change 
in a given proposition in consequence of which a second 
proposition is obtained, which cofitains a double negation 
of the original one; thus, "No just acts are not expedi- 
ent" is a double negation of the proposition "All just 
acts are expedient," and it is therefore the obverse of 
that proposition. A proposition is obverted by putting 
into it a double negation. 

This double negation is effected, either by using a 
negative word in each of the terms of the proposition, 
or by placing both negatives in the predicate part of 
the proposition; thus, the proposition, "All just acts 
are honorable," is obverted by placing a negative in 
both its terms, and the obverted proposition becomes, 
"No just acts are not honorable." The proposition, 
"Some just acts are expedient," is obverted by put- 
ting the double negation into the predicate part ; and 
the obverted proposition is, "Some just «cts are not 
not-expedient," — not inexpedient things. It should 
be noted that when a negative proposition is obverted 
it becomes aflfirmative in quahty. Thus, "No men are 
always happy" obverted, becomes "All men are not- 
happy beings." 



92 ELEMENTARY LOGIC 

Let the student carefully note the difference in the 
negatives in these two propositions : In the proposition, 
"No men are always happy," it is the assertion that 
is negative, the proposition is equivalent to the follow- 
ing, "No men are any part of the class of happy beings; " 
in the proposition "All men are not happy beings" it 
is the predicate term that is negated, and this is done 
by prefixing the negative word to the predicate term, 
joining the two by a hyphen. This proposition is 
then equivalent to the following, "All men are in the 
class of not-always-happy beings." The student must 
keep in mind this difference between negating an asser- 
tion and negating the terms of a proposition. Obver- 
sion is based upon the principle that two negatives 
are in effect an affirmative. The particular negative 
proposition is ob verted by joining the negative to the 
predicate term. 

The peculiarity of obversion is that it changes the 
quahty of a proposition, but so as to express the same 
judgment. The student should note that in obvert- 
ing the particular negative proposition, the obverted 
proposition contains but one negative, and that is 
joined to the predicate term. Thus, obverting "Some 
men are not honest," we get "Some men are not-honest 
beings, " or "Some men are dishonest." Note also that 
the quahty of this proposition is affirmative ; it means 
"Some men are in the class of not-honest, or dishonest, 
beings"; whereas the original proposition is negative. 

The following directions may be of service to the 
student in obverting propositions: — 



DEDUCTIVE INFERENCE 93 

(i) Obvert a universal affirmative proposition by 
using a negative in both its terms. 

(2) Obvert a universal negative by giving the sub- 
ject term the form it has in the universal affirmative, 
and join the negative word to the predicate term. 

(3) Obvert a particular affirmative by using two 
negatives in the predicate, joining one of them to the 
predicate term. 

(4) Obvert a particular negative by joining the 
negative word to the predicate term. 

2. Inference by Conversion, — By conversion is 
meant a change in the position of the subject and 
predicate terms of a proposition, which gives a second 
proposition, having the same quaUty as the original 
proposition, but in which the subject and predicate 
terms of the original proposition have exchanged places. 
Thus, to convert the proposition, "Some men are 
happy," I change the position of the subject and predi- 
cate terms, making them exchange places, as it were, 
and thus obtain the proposition, "Some happy (beings) 
are some men," which expresses the same judgment 
as the original proposition. The student must bear 
in mind that conversion, unhke obversion, does not 
change the quahty of a proposition. 

Logicians distinguish three varieties of conversion, or 
three methods of effecting it, — conversion by limita- 
tion, or per accidens, simple conversion, and conversion 
by negation or by contraposition. This last species of 
conversion must, I maintain, be rejected, because, as 
I shall show, it involves a change in the quality of the 



94 ELEMENTARY LOGIC 

proposition. Conversion by limitation applies to the 
universal affirmative proposition; and it consists in 
limiting the extension of the new subject term. Thus, 
to convert the proposition, "All men love life, "I must not 
only exchange the positions of the subject and predicate 
terms, but the new subject term must be hmited in its 
extension. 

The converted proposition, therefore, is, "Some 
lovers of life are all men." Did I not make this change 
in the subject term, the two propositions would not 
be equipollent; for the original proposition means 
"All men are some of those beings that love life " ; and 
to convert the original proposition without limiting the 
extension of the new subject term would give," All lovers 
of life are all men," which may be a true proposition, 
but which does not certainly follow from the original 
proposition, nor does it express the same judgment as 
the original proposition. 

Simple conversion is effected by exchanging the posi- 
tions of the subject and predicate terms. This form 
of conversion apphes to the particular affirmative 
proposition. Thus, the proposition, "Some A is B," 
converted, becomes, "Some B is A.'" The surest and 
easiest way of performing conversion is, first to quantify 
the given proposition, and then convert simply. For 
example, suppose we have the proposition, "A stitch 
in time saves nine ' ' ; first, quantifying this proposition 
we get, "A stitch in time is one of the things, or is 
some part of that which saves nine," then by convert- 
ing simply, we obtain, "One of the things that save 



DEDUCTIVE INFERENCE 95 

nine is a stitch in time," or, " Some part of that which 
saves nine is a stitch in time." 

I advise the student to follow in practice this one 
direction in converting all propositions, — quantify the 
given proposition and then exchange the positions of 
the subject and predicate terms. 

3. Inference by Contraposition. — The contrapositive 
of a term is that term negated ; the contrapositive of a 
proposition is a proposition which is obtained by ob- 
verting and then converting a given proposition. Thus, 
the contrapositive to the proposition, "All men love 
happiness," is, "Those who do not love happiness are 
not men." Hence, to obtain the contrapositive to any 
proposition, the simple direction is, first obvert the propo- 
sition, then convert this obverted proposition. 

We have seen that the particular negative proposition 
cannot be converted, since to do so involves a change 
in its quahty, which the principle of conversion does 
not permit. For example, were I to convert the propo- 
sition "Some men are not honest," according to the 
method of negation, I should obtain, "Some not-honest 
beings are some men, " which is an affirmative propo- 
sition. The contrapositive to this proposition, how- 
ever, can be given. Thus, obverting, it becomes, 
" Some men are some not-honest beings," and this con- 
verted gives, "Some not-honest beings are some men." 

4. Inference by Added Determinants. — This mode 
of inference is effected by adding the same quahfying 
word to the terms of the given proposition. Thus, 
from the proposition, "All metals are elements," I can 



96 ELEMENTARY LOGIC 

infer that all heavy metals are heavy elements. But 
from the proposition, "Ants are animals," it would be 
wrong to infer that large ants are large animals. 
Hence, a caution must be observed in the use of added 
determinants. The added determinants must have 
the same meaning in both terms. This caution is not 
heeded when one infers from the proposition, "Poets 
are men, " that bad poets are bad men. That which 
makes a man a bad poet does not necessarily make him 
bad as a man. 

Section 21 
inference by opposition 

We have seen what are the modes of inference which 
are based upon the relation of equipollence. Let us 
now examine the methods of inference which are based 
upon opposition, the other relation on which immediate 
inference is founded. 

Two propositions may be opposed to each other in 
four ways : — 

(i) As contraries, 

(2) as contradictories, 

(3) as subcontraries, 

(4) as one subaltern to the other. 

The strongest form of opposition is that between con- 
traries ; the least degree of opposition is that between a 
subaltern and the universal over it. The contrary re- 
lation is that between the universal affirmative and the 
universal negative, or that between two universals of 
opposite quahty. 



DEDUCTIVE INFERENCE 97 

The contradictory relation is that between a universal 
of one quality and a particular proposition of the oppo- 
site quahty; that is, between the universal affirmative 
and the particular negative, and between the universal 
negative and the particular affirmative. There are, 
therefore, two contradictories and one contrary. 

The subcontrary relation obtains between the par- 
ticular affirmative and the particular negative proposi- 
tions, and the subaltern relation holds between each 
universal and the particular of the same quahty. The 
following are examples of these various forms of oppo- 
sition between propositions: The propositions, "All A 
is B" "No A is B," are contrary to each other. "All 
A is 5," "Some A is not B," contradict each other; and 
so do "No A is 5," and "Some A is B. " 

"Some A is 5," and "Some A is not 5," are sub- 
contraries. 

"Some ^ is 5" is the subaltern of "All ^ is 5," and 
"Some A is not B" is the subakern of "No A is 5." 

In formal logic it is customary to represent these 
forms of opposition by the follow- 
ing diagram : — 

In this diagram the letters are 
symbols of the propositions. 

A the symbol for the universal 
affirmative. 

E the symbol for the universal 
negative. 

/ the symbol for the particular affirmative. 

O the symbol for the particular negative. 




g8 ELEMENTARY LOGIC 

The upper side of the square represents the contrary 
relation; the diagonals of the square stand for the 
two contradictories; the lower side of the square, 
for the subcontrary relation; and the subalterns are 
represented by the two remaining sides of the square. 
It will be readily seen from this diagram that the con- 
trary relation is that between A and E propositions; 
the contradictory is that between A and O, and between 
E and /. The subcontrary is that between / and O ; 
and the subaltern that between A and /, and between 
E and O. 

The student should accustom himself to the use of 
these symbols, and to the expression of the quantity 
and quality of propositions, and the various kinds 
of opposition between them, in term of these sym- 
bols. 

I. Now, let us see what inferences are possible, taking 
any one of these propositions as a datum. We will 
begin with the contrary relation. The peculiarity of the 
contrary is this : if the given proposition is a true propo- 
sition, we can certainly infer that its contrary is false; 
but if the given proposition is not true, we cannot be 
certain that its contrary is true. For example, given 
the true proposition, "All men love happiness," I am 
certain that the proposition, "No men love happiness," 
is false ; but if this first proposition is not true, I cannot 
be certain whether its contrary is true or false. The 
peculiarity of the relation between contraries is, there- 
fore, that both cannot be true, but both can be false. 
Hence, the relation of contrary allows but one inference, 



DEDUCTIVE INFERENCE 99 

the inference that one of the contraries is false if the 
other proposition is true. 

2. Examination of the contradictory relation will 
show that it permits two inferences; the peculiarity 
of this relation being, that if one of these propositions 
is true, the other is false, and conversely, if one is false, 
the other is true. Hence, two inferences are possible. 
Let me explain by an example. Let the given propo- 
sition be, "All men love happiness," and assume that 
this proposition is true; then it certainly follows that 
the proposition, "Some men do not love happiness," 
is false. Now, assume that the first proposition is 
false, then it will as certainly follow that the proposi- 
tion, "Some men do not love happiness" is true. It 
cannot be false to assert that all men love happiness, 
unless it is true that some men do not love happi- 
ness. 

3. The subcontrary relation makes it possible for 
both propositions to be true, but not possible for both 
to be false; thus, the propositions, "Some men are 
honest," and "Some men are not honest," can both 
be true; but they cannot both be false. The reason 
for this is obvious; take the first proposition, "Some 
men are honest," and, this being by supposition false, 
its contradictory, "No men are honest," must be true; 
and if so, the particular proposition, "Some men are 
not honest," must be true also. Hence, the subcon- 
trary relation permits one inference only, the inference 
from the falsity of one of the subcontraries to the truth 
of the other. , ^, 

tOFCr 



100 ELEMENTARY LOGIC 

4. Finally, the subaltern relation makes possible 
two inferences : the truth of the subalterns from the 
truth of the universals over them, and the truth of the 
universals from the falsity of their subalterns. For 
example, from the proposition, "All ^'s are 5's, " I 
am certain that some A's are 5's; and likewise from 
the proposition, "No ^'s are jB's, " I am certain that 
some ^'s are not 5's. But if these universals are false, 
I cannot be certain whether their subalterns are false 
or true; nor can I be certain that these universals are 
true, if their subalterns are true. Because some A is 
B it does not necessarily follow that all A is B. But 
let the subaltern be false, then will the universal over it 
be also false. If it is not true that some men are honest, 
it cannot be true that all men are honest. 



CHAPTER VIII 

MEDIATE INFERENCE, THE SYLLOGISM 

The syllogism, or mediate inference as the term 
implies, is inference by means of some intermediary 
conception or judgment. 

Section 22 
description of the syllogism 

The form of mediate inference is the Syllogism, 
which is a combination of two propositions from which 
a third proposition necessarily follows. The syllogism, 
accordingly, consists of three propositions, two of which 
are called the premises, and the third the conclusion. 
I shall first describe the syllogism in its regular forms. 
Of regular syllogisms there are three kinds: — 

(i) the categorical, 

(2) the hypothetical, and 

(3) the disjunctive. I shall describe these in the order 
named. 

I. A categorical syllogism is a syllogism the prem- 
ises of which are categorical propositions. The 
syllogism contains three terms, called the major, the 
minor, and the middle terms. The major term is the 
term of greatest extension ; and the premise in which it 



I02 ELEMENTARY LOGIC 

occurs is therefore called the major premise. The 
minor term is the term of lesser extension; and the 
premise in which it occurs is called the minor premise. 
The middle term is so called from its function, which 
is that of uniting the two other terms; it is by means 
of this term that the major and minor terms are united 
in the conclusion; the middle term is therefore the 
connecting link between the other two terms. 

The structure of the categorical syllogism can best be 
understood by an example. "All metals are elements; 
iron is a metal, therefore iron is an element." In this 
syllogism, the major term is element, since this is the 
term of largest extension; iron is the minor term; 
because it is, compared with the term element, a term 
of lesser extension; and the middle term is metal, 
because it is the term by means of which the minor and 
major terms are connected in the conclusion. The 
major premise in this syllogism is, "metals are ele- 
ments," because this proposition contains the major 
term; and the minor premise is, "iron is metal," 
because this proposition contains the minor term. It 
should be noted also that the middle term, metal, 
occurs in each of the premises; and that both major 
and minor terms occur in the conclusion. 

Figures of the Syllogism. — I have described the 
general structure of the categorical syllogism. I must 
next describe certain forms which this syllogism as- 
sumes according to the position and combination of 
its terms; these are the so-called Figures of the syllo- 
gism. The figure of the syllogism is the form which 



MEDIATE INFERENCE, THE SYLLOGISM 103 

is determined by the position of the middle term in 
the premises relative to the other two terms. Exam- 
ination of a syllogism will make it evident that four 
arrangements of the middle term with the other two 
are possible: — 

(i) the middle term can be the subject in the major 
premise, and the predicate in the minor premise ; 

(2) the middle term can be the predicate in both 
premises ; 

(3) the middle term can be the subject in both 
premises ; 

(4) the middle term can be the predicate in the 
first premise, and the subject in the second premise. 

Using symbols for the three terms, P for major, 
5 for minor, and M for middle term, we can repre- 
sent these four arrangements of the terms and proposi- 
tions as follows : — 

Fig. I. 



Fig. III. 



M — P 


Fig. II. 


P — M 


S — M 




S — M 


S — P 


S — P 


M — P 


Fig. IV. 


S — M 


M — S 




M — P 



These diagrams represent the syllogistic figures usually 
designated as Fig. I, Fig. II, Fig. Ill, and Fig. IV. 
It will be seen that the characteristics of these figures 



I04 ELEMENTARY LOGIC 

are the following: In Fig. I, the middle term is the 
subject in the major premise, and the predicate in the 
minor premise. In Fig. II, the middle term is the predi- 
cate in both premises. In Fig. Ill, it is the subject in 
both premises. In Fig. IV, the position of the middle 
term is the same as its position in Fig. I, the only dif- 
ference between these figures being, that in Fig. IV 
the order of the premises is reversed, the minor prem- 
ise being placed first. Figure I was the typical syllo- 
gism of Aristotle, though he recognizes Figs. II and 
III, but as imperfect forms of the syllogism. 

Moods in the Syllogism. — Mood is the form of the 
syllogism which is determined by the quantity and 
quality of the propositions that form its premises. 
Now, since there are four such propositions, distin- 
guished by their quantity and quality, there can be as 
many moods in each of the syllogistic figures as there 
are different possible combinations of the four propo- 
sitions, taken two at a time. For example, in Fig. I, 
there can be the following commnations: the univer- 
sal affirmative or A proposition -may be used for both 
premises ; the universal negative or 2^ proposition, the 
particular affirmative or / proposition, and the particu- 
lar negative or O proposition can likewise be used. 
Each of these universals may be cSwibined in two 
ways with each of the other propositions; and the 
same is true of each of the particular propositions. 
The following arrangements show the total number 
of such combinations, and consequently the total 
number of moods possible in Fig. I : — 



MEDIATE INFERENCE, THE SYLLOGISM 105 



I 


2 


3 


4 


A A A A 


EEEE 


1 1 1 1 





A E I 


AEI 


AEIO 


A I EO 



Here are four groups, formed by combining each of the 
four propositions with each of the other propositions; 
and each group gives four combinations, the total num- 
ber being sixteen; that is, there are sixteen possible 
moods in Fig. I. As each figure gives the same number 
of such moods, the total number of possible moods of 
the syllogism is sixty-four. But, as will be explained 
later, only about one fourth of these moods are valid 
syllogisms, that is, syllogisms which yield a conclusion. 
2. The Hypothetical Syllogism. — A hypothetical 
syllogism is a syllogism the major or first premise of 
which is a hypothetical proposition, and the minor 
premise a categorical proposition. In this syllogism the 
major premise states a supposition or condition and its 
consequence or result. The minor premise asserts either 
the truth or the untruth of the supposition, or it asserts 
the same of the consequence stated in the major premise. 
The conclusion, according to th^ assertion made by the 
minor premise, affirms or denies either the supposition 
or the consequence. When this conclusion is an affirm- 
ative proposition, the syllogism is called constructive; 
when it is a negative proposition, the Syllogism is called 
destructive. 
The minor premise can make four assertions: — 
(i) that the supposition made in the major premise 
is true, 



I06 ELEMENTARY LOGIC 

(2) that this supposition is not true, 

(3) that the consequence or result stated in the 
major premise is fact, or 

(4) that this consequence is not fact. 

We shall later see that in a valid hypothetical syllo- 
gism the minor can make but two assertions. These 
features of the hypothetical syllogism are exhibited in 
the following examples : — 

1. li A is B, Cis D; 

A is B; 
Therefore, C is D. 

2. li A is B, C is D; 

C is not D ; 
Therefore, A is not B. 

3. li A is B; C is D; 

A is not B; 
No conclusion. 

4. li A is B, C is D; 

CisD; 

No conclusion. 

5. If A is not B, C is not D ; 

A is not B; 
Therefore, C is not D. 

6. If A is not B, C is D; 

A is not B; 
Therefore, C is D. 



MEDIATE INFERENCE, THE SYLLOGISM 107 

Two peculiarities of the hypothetical syllogism are 
shown in examples 5 and 6. In 5, although the minor 
premise is a negative proposition in form, it asserts the 
truth of the supposition made in the major premise, viz. 
that A is not B. In 6, an affirmative conclusion is 
dravm from what appears to be a negative premise ; in 
reality, however, the major premise is an affirmative 
proposition, the negation in it not aflfecting the asser- 
tion. 

3. The Disjunctive Syllogism. — A disjunctive syl- 
logism is one having for its major premise a disjunctive 
proposition, and for its minor premise a categorical 
proposition, the conclusion of course being a categori- 
cal proposition. We saw that the function of the dis- 
junctive judgment is to present alternatives. Accord- 
ingly, the major premise of this syllogism presents 
two or more alternatives; the minor premise accepts 
or rejects one or more of these alternatives; and the 
conclusion is affirmative or negative, according to the 
character of the minor premise. Logicians distin- 
guish two moods of this syllogism, technically desig- 
nated, modus ponendo tollens and modus tollendo 
ponens. These words are determined by the character 
of the minor premise; if the minor rejects or takes 
away an alternative, the conclusion is affirmative; 
we have then the modus tollendo ponens; if, on the 
contrary, it accepts an alternative, the conclusion is 
negative; and we have, therefore, the modus ponendo 
tollens. The following syllogisms illustrate these two 
moods : — 



I08 ELEMENTARY LOGIC 

A is either B ox C\ 

A is not C; 
Therefore, it is B. 

In this syllogism, since the. minor rejects one alterna- 
tive, the other alternative is affirmed; we have, there- 
fore, a case of modus tollendo ponens. 

A is either B or C\ 
A'lsB; 

It is therefore not C. 

Hence, we have the modus ponendo tollens. 

Irregular Syllogisms. — Having described the syllo- 
gism in all its regular forms, I will now describe those 
deviations from the regular syllogisms which are very 
common in our reasonings. 

These irregularities in syllogistic reasoning are pro- 
duced in two ways : — 

(i) by combining features of the syllogism in its 
different forms; 

(2) by abridgment or condensation in a syllogistic 
argument, this abridgment being made either by omit- 
ting one of the propositions of a single syllogism, or 
by combining several syllogisms so as to form a chain 
of reasoning. 

The irregular forms of reasoning thus produced are 
the following : — 

(i) the Dilemma, 

(2) The Enthymeme, and 

(3) the Sorites. These shall be described in their 
order. 



MEDIATE INFERENCE, THE SYLLOGISM 109 

I. The Dilemma. — The dilemma is a form of ar- 
gument in which two or more alternatives are so pre- 
sented that a certain conclusion is inevitable, whichever 
of these alternatives is accepted or rejected by a dis- 
putant. These alternatives are the so-called horns of 
the dilemma ; one or the other has to be taken v^dth a 
damaging, if not fatal, result in either case. The form 
in which this sort of argument is presented is a syllo- 
gism having for its major premise a h^'pothetical propo- 
sition which presents at least two alternatives, and 
for its minor premise a disjunctive proposition; the 
conclusion of this syllogism is either a categorical or a 
disjunctive proposition. 

This peculiar structure will be understood best by 
an example. Take the following argument: "Every 
law is either useless, or it occasions hurt to some person. 
Now, a law that is useless ought to be abolished, and 
so ought every law that occasions hurt; therefore, 
every law ought to be abolished." The argument is a 
dilemma. We have only slightly to change its form 
and we shall get a dilemmic syllogism. The major 
premise is, "If a law is useless it ought to be abolished, 
and if a law is harmful it ought to be abolished." 
The minor premise is, "Every law is either useless 
or harmful." The conclusion is, "Every law ought 
to be abolished." 

Here is a typical dilemma; two alternatives are 
offered in the major premise, one or the other of 
these is accepted in the minor, and a certain con- 
clusion follows. The two horns of this dilemma are 



1 10 ' ELEMENTARY LOGIC 

the uselessness of law and the harmfulness of law. 
Let either of these be accepted, and the same con- 
clusion follows. Of course, the escape from this 
dilemma lies in challenging the major premise, in 
rejecting the alternatives, either on the ground that 
they contain unwarranted assumptions, or that they do 
not exhaust the possible alternatives. But this matter 
belongs to another chapter. 

The dilemma is called constructive when it leads 
to an affirmative conclusion, and destructive when the 
conclusion established is negative. 

Dilemmas are also simple or complex, according 
as the conclusion is a categorical or a disjunctive 
proposition. 

I give the following examples which will make clear 
these forms of the dilemma : — 

1. Simple constructive dilemma : — 

li A \s B, C \& D; and if £ is F, C is Z>; but 
either Ais B or Eis F] therefore, C is D. 

2. A complex constructive dilemma : — 

If A\sB,C is D; and if E is F, G is H; but 
either A is B, or E is F ; therefore, either C is 
D, or G is H, 

3. A simple destructive dilemma : — 

li Ais B, Cis D; and if A is B, E is F; but 
either C is not D, or E is not F ; therefore, A is 
not 5. 



MEDIATE INFERENCE, THE SYLLOGISM III 

4. A complex destructive dilemma : — 

li A is B,C is D; and ii E is F,G is H; but 
either C is not D, or G is not H ; therefore, A is 
not B, or E is not F. 

Jevons, Whately, Mansel, and some other logicians 
reject the simple destructive dilemma, but without 
sufficient reason, I think. These logicians, I should 
add, teach that in the dilemma two distinct antece- 
dents as well as two alternatives must be presented. Of 
course if their conception of the dilemma is the correct 
one, there cannot be a simple destructive dilemma, 
since such a dilemma does not have two distinct ante- 
cedents. I maintain, however, that the description of 
the dilemma given above is correct; and if so, the 
simple destructive dilemma is as admissible as is the 
complex destructive dilemma. 

2. The Enthymeme. — An enthymeme is a syllogism 
with one of its propositions omitted. Any one of the 
three propositions may be the omitted one; but it is 
more common to omit one of the premises. The fol- 
lowing will serve as examples of enthymemes : — 

1. The English government is liable to frequent changes 

in its foreign pohcy, because it is democratic. 

2. The EngHsh government is hable to frequent changes 

in its foreign policy, because all democratic gov- 
ernments are hable to such changes. 

3. All democratic governments are hable to frequent 

changes in their foreign pohcies ; and the Enghsh 
government is democratic. 



112 ELEMENTARY LOGIC 

Inspection of these examples discovers that in i, it is 
the major premise that is omitted; supplying it, the 
complete syllogism is, "All democratic governments 
are liable to frequent changes in their foreign policies ; 
the Enghsh government is democratic; therefore, the 
English government is liable to frequent changes in 
its foreign policy." In example 2, the student will per- 
ceive that a minor premise is to be supphed; and in 
the third example it is the conclusion which is not stated. 
The enthymeme, though an irregular construction, 
is the most common form in which deductive reason- 
ing occurs, the complete syllogism being rarely met with 
outside of text-books on logic. 

3. The Sorites. — A chain of syllogisms is a series 
so constructed, that the conclusions of one syl- 
logism, either expressed or implied, form a premise 
for the next. Prosyllogism is the name given to a 
syllogism that proves the premises of the following syllo- 
gism; and episyllogism is the name given to a syllo- 
gism that follows and rests back upon a preceding one 
for its support. The syllogistic chain assumes two 
forms, determined by the way in which the syllogisms 
are combined to form the given chain. The first of 
these is the epichirema, of which the following is an 
example: "All M is P because M is X; All 5 is M 
because 5 is F; therefore. All S is P." If we examine 
this peculiar combination of propositions, we shall find 
that they constitute a chain of syllogisms; for each 
premise in this single syllogism is an enthymeme, and 
the enthymemes expanded give the following chain : — 



MEDIATE INFERENCE, THE SYLLOGISM 1 13 

Whatever is X is P; 
M is X; 
Therefore, M is P. 

Whatever is F is M; 
5 is F; 
Therefore, S is M. 

Whatever is If is- P; 
S is M\ 
Therefore, 5 is P. 

The epichirema can, t-herefore, be described as a 
syllogism with supporting reasons for its premises, or 
as a syllogism whose premises are enthymemes. The 
more typical form which the syllogistic chain presents 
is the sorites, which may be defined as a chain of en- 
thymemes. 

There are two well-known forms of sorites, one 
called the Aristotelian, the other the Goclenian, from 
Goclenius, a German logician in the seventeenth cen- 
tury. The following is an example of the Aristotelian 
sorites : — 

All^ is 5; 

All J5isC; 

All C isD, 

AMD is E, 

All E is P; 

Therefore, All A is P. 

The student will readily see that this chain is formed 
by combining syllogisms of Fig. IV, by omitting the 



114 ELEMENTARY LOGIC 

conclusion of each prosyllogism. If the omitted propo- 
sitions are suppUed, the chain becomes the following : — 

A is 5; 

BisC; 
Therefore, -4 is C 

A isC; 

CisD; 
Therefore, A is D. 

AisD; 

DisE; 
Therefore, A is E. 

It will be seen that in this chain the unexpressed con- 
clusion of each syllogism is the minor premise of the 
succeeding syllogism; and that the subject of the first 
proposition is also the subject of the last proposition 
in the sorites. 
The following is an example of the Goclenian sorites : 

A is B; 
CisA; 
DisC; 
EisD; 
FisE; 
Therefore, F is B. 

The student should see in this chain a combination 
of syllogisms in Fig. I, in which the unexpressed con- 
clusion of each prosyllogism is the implied minor 
premise of the syllogism following it. Another thing 
should be also seen : this sorites, if its propositions are 



MEDIATE INFERENCE, THE SYLLOGISM 115 

read in the reverse or backward order, is the same as 
the Aristotehan sorites. Thus, beginning with the 
last proposition but one, the chain becomes — 

F is E; 
E is D] 
D is C; 
Cis ^; 
Therefore, F is A. 

which is our own Aristotehan sorites. The real dis- 
tinction between these two sorites is, that in the Aris- 
totehan sorites the reasoning is progressive, while in 
the Goclenian sorites the reasoning is retrogressive. 

Section 23 

regulative principles and rules for the 
syllogism 

The syllogism has now been described in all its forms 
and varieties; it remains to ascertain the principles 
of valid reasoning, and the rules which must be 
observed in the employment of the syllogism. 

A syllogism is vahd when the conclusion follows 
from the given premises, irrespective of the truth or 
untruth of the premises in themselves considered. It 
is not, therefore, essential to the vahdity of a syllogism 
that its conclusion be a true proposition; but only 
that its conclusion follow from, and be consistent with, 
the premises; a valid syllogism being one which gives 
for its conclusion a true proposition if the premises 



Il6 ELEMENTARY LOGIC 

are true propositions, and a false proposition if its 
premises are not true. It should also be borne in 
mind that it is not the function of formal logic to find 
or to establish true premises; consistency, not truth, 
being, as has been said, the aim of this part of logic. 
The Validity of Disjunctive and Hypothetical Syl- 
logisms. — In examining the conditions and the rules 
of valid syllogisms, we can best begin with those which 
are of simplest construction, the disjunctive and the 
hypothetical syllogisms. 

1. The disjunctive syllogism is based upon the prin- 
ciple of alternatives, which, as we have seen, it is the 
function of the disjunctive judgment to present. This 
principle requires that, if one or more of the given 
alternatives are accepted, the remaining alternative 
or alternatives must be rejected. From this principle 
we derive the following rule for the disjunctive syllo- 
gism: The minor premise must either affirm or deny 
some one of the given alternatives. 

2. The hypothetical syllogism is based upon the 
principle of supposition and consequence, or antece- 
dent condition and consequent. This principle is, 
that if the supposition is fact, or is true, the consequence 
is also fact, or is true; or, otherwise expressed, if a 
specified condition is fulfilled, its consequence is fact. 
We can therefore always infer the fact or reality of a 
specified consequence from the truth of the supposition 
or from the fulfillment of the condition. But this 
principle does not permit us to infer the truth of a sup- 
position, or of a condition, from the truth or fact of 



MEDIATE INFERENCE, THE SYLLOGISM I17 

a given consequence; because, while the thing which 
is called a consequent must exist if its supposed condi- 
tion exists, this thing might conceivably exist if that 
particular condition did not exist; hence, the mere 
fact of this consequence proves nothing. 

This will be made clearer by an example. Take 
the following, "If it is cloudy, there will be no dew 
to-night." Now, suppose that we learn that it is 
cloudy, we shall then be certain that there will be 
no dew; but suppose we are told to-morrow morning 
that there is no dew, can we be certain that it was a 
cloudy night? Not unless we know that a cloudy 
night is the only condition of there being no dew; 
and the proposition does not tell us that. But now, 
suppose that we are told that there was dew; then 
we can be certain that the night was not cloudy. 
Hence, the principle of supposition permits us to infer 
from the nonexistence of the consequence, the nonexist- 
ence of the condition, or supposed fact. Accordingly, 
we derive the following rule for the hypothetical syl- 
logism in its usual form, The minor premise must either 
afhrm the condition, or deny the consequent. 

There is, however, a permissible form of this syl- 
logism which gives a conclusion when the minor 
premise affirms the consequent. The following is a 
case of this sort, "Only if A is honest can he gain 
my confidence; he has gained my confidence; can 
you not infer that A is honest?" The peculiarity of 
this syllogism is that the major premise states an ex- 
clusive hypothesis; the honesty of A is the sole thing, 



Il8 ELEMENTARY LOGIC 

which, if it exist, can give me confidence in him ; and 
consequently if it is fact that A has gained my confi- 
dence, it must be fact that he is honest. 

Our rule for the hypothetical syllogism must accept 
this quahfication, viz., When the major premise contains 
an exclusive hypothesis, the minor premise can also 
affirm the consequent. 

3. The regulative principles and rules for the cate- 
gorical syllogism can be most easily defined if we treat 
the syllogistic propositions as if they assert a relation 
between two classes, or the relation of part to whole; 
in fact, the rules laid down in most text-books are based 
upon this assumption. 

We have seen that in the syllogism, a connection of 
some sort, established between the middle term and the 
major term in one premise, and between the middle 
term and the minor in the other premise, makes the 
conclusion of the syllogism necessary. Our problem, 
therefore, is to determine what this connection is, and 
how we can be certain when this connection between 
the terms has been established. 

Now, if we assume that the syllogistic inference is 
based upon the class relation, the principle on which 
that inference proceeds is that of inclusion in, or exclu- 
sion from, a class. Thus, to take the classic syllo- 
gism : — 

All men are mortal; 
Caius is a man; 
Therefore, Caius is mortal. 



MEDIATE INFERENCE, THE SYLLOGISM II9 

Caius is included in the class man, and this class is 
included in the class mortal beings ; hence, Caius 
also must be included in that class. 

Logicians distinguish two sets of rules for the cate- 
gorical syllogism: general rules, those which apply- 
to any syllogism ; and special rules, those which apply 
to the syllogism in each of the four figures. We shall 
follow this distinction; and, accordingly, let us first 
ascertain the general rules and the reasons for them. 
One such rule is, that the middle term must be uni- 
versal in one premise at least. The reason for this 
rule is, that were this term taken only in part of its ex- 
tension in both premises, it could establish no certain 
connection between the major and minor terms. A 
glance at the syllogism will make this evident. Sup- 
pose the premises to be the following : — 

Some A is B] 
All C is A. 

Let us quantify the propositions, and the premises 
become, " Some of A is some part of B ; All of C is some 
part of A." Now it is obvious, that with these premises 
no connection is necessarily established between C and 
B; for, although C is in yl, it is not necessarily in that 
part of A which is included in, or subsumed in B. Let 
us test this rule by a syllogism in a different figure, say 
Fig. IL 

All A is some part of B; 
All C is some part of B; 



120 ELEMENTARY LOGIC 

In this syllogism the middle term B is not taken in its 
full extension; and the consequence is it cannot be 
determined whether A and C are connected or not; 
since they can both be a part of B, without either being 
a part of the other. Two circles can be put within the 
same circle without necessarily intersecting each other. 
The student who is so minded can test this rule in the 
other two figures of the syllogism; and he will find 
the use of the circles as in quantification a serviceable 
method. We can formulate this first general rule 
of the categorical syllogism in this way, One premise 
at least must be universal. 

The second general rule of the syllogism is, One prem- 
ise at least must be affirmative. The reason for this 
rule is, that if both premises are negative, the relation 
between the middle term and the other two terms being 
one of exclusion, no certain connection can be estab- 
lished between those terms. It is just as when two 
circles are outside the same circle; it cannot be deter- 
mined in that case whether those circles are outside 
of each other or whether they intersect. 

The third general rule in the syllogism is, that if one 
premise is particular the conclusion can only be par- 
ticular. Let the premises be: — 

All A \s B; or All A is some part of B ; 
Some C is A; Some C is some part of A . 

It is obvious that with these premises, we can only 
be certain of the part of C that is included in the middle 
term A . Let the premises be : — 



MEDIATE INFERENCE, THE SYLLOGISM 121 

All A is some part of B ; 
Some C is not any part of B. 

It can only be certain in this case that some C is no part 
of A. 

The fourth and last general rule is, If one premise 
is negative, the conclusion is negative. Thus, if the 
premises are — 

All A is B; 
No C is B; 

it is obvious that, since C is excluded from B, in which 
A is included, the relation between C and A is one of 
exclusion, and that is a negative relation. 

Now, it might be supposed that any syllogism which 
conformed to these general rules would be valid; 
but examination will show that such is not the case. 
Take, for instance, these premises : — 

AW A isB; 
No C is A. 

These premises do not violate any general rule of the 
syllogism. Why not, then, draw the conclusion that 
No C is ^ ? The reason will be at once plain if we 
note what the minor premise does; it excludes the 
class C from the class A ; and the major premise having 
included this A class in the class B, we cannot be cer- 
tain that C is also included in B or is excluded from B. 



122 ELEMENTARY LOGIC 

The following arrangement of the circles shows this 
ambiguous position of C: — 




The circle C can be in the B circle, or outside that 
circle ; and in either position be disconnected from the 
A circle. 

Now, observe that these premises from which no 
conclusion follows are the premises of the syllogism 
in Fig. I ; and that it is the minor premise that is nega- 
tive. Hence, one condition of a valid syllogism in 
Fig. I is, that the minor premise must be affirmative. 

Let us next inspect the following premises, also in Fig. I. 

Some A'\^ B\ 
All C is A. 

Why not conclude. Some C is 5? The reason 
evidently is, that the major premise being a particular 
proposition, only a part of the middle term A is in the 
major term B\ and, although the minor premise in- 
cludes C in vl, it does not necessarily include it in that 
part of A which is in B, as the circles will show. 




In this diagram the position of the circle C is ambigu- 
ous. Hence, a second special condition of a valid 



MEDIATE INFERENCE, THE SYLLOGISM 1 23 

syllogism in Fig. I is, that the major premise must be 
universal. If we unite these two special conditions, we 
get as the first of the special rules of the syllogism the 
following; In Fig. I the major premise must be uni- 
versal, and the minor premise must be affirmative. 

We will next examine the premises of a syllogism 
in Fig. II. 

Alibis 5; 
All C is B. 

Why should not these premises give a conclusion as 
these propositions would do in Fig. I? The obvious 
reason is, that these premises violate the general rule 
which requires that the middle term should be taken 
in its fullest extension in one premise at least. Now, 
if we quantify these propositions, it will be clearly seen 
that the middle term B is not taken in its full extension. 
The use of the circles will make this fact apparent. 
Two circles can be placed within the same circle with- 
out necessarily intersecting. Hence, the second special 
rule of the syllogism is. In Fig. II one premise must 
be negative. 

We observe also that the syllogism in Fig. II has 
this pecuharity, that it gives only negative conclusions. 

We will next see what special rule, if any, is required 
for Fig. HI. Take the following premises : — 

A is B\ 
A is not C. 



124 ELEMENTARY LOGIC 

Will these premises give a conclusion? Quantified, 
they read: — 

All of A is some part of B; 
None of A is any part of C 

Representing these premises by circles, we get the 
following : — 




From this it appears that the circle C can occupy either 
of two positions and remain outside the circle A. 
Notice it is the minor premise that is negative, and 
yields this ambiguous result. Hence, we conclude 
that in Fig. Ill as in Fig. I the minor premise cannot 
be negative. 
Take again, in the same figure, these premises: — 

All A is some part of B; 
All A is some part of C. 

Since both these premises are universal propositions, 
we might expect that the conclusion would be universal 
also, as it certainly is in the other figures we have exam- 
ined ; but examination will show that only a particular 
conclusion is admissible with these premises. Again, 



MEDIATE INFERENCE, THE SYLLOGISM 125 

let the circles be drawn and the truth of this statement 
becomes clear. 




The circle A is within two circles and these circles 
need have only a part of their areas in common. Put- 
ting together these two results, we get the third special 
rule of the syllogism : In Fig. Ill the minor premise 
must be affirmative, and the conclusion only can be 
particular. 

Finally, there remains to be examined the syllogism 
in Fig. IV. This figure, as we have seen, diflfers from 
Fig. I only in having the minor premise first. Now, 
examination of the syllogism in this figure shows that 
the minor premise cannot be negative. But it also 
appears that in Fig. IV the major premise cannot be 
particular. We get the following rule, which apphes 
to syllogisms in Fig. IV : The minor premise cannot 
be negative and the major premise cannot be par- 
ticular. 

These special rules, we have ascertained and proved, 
can be reduced to three; since some of them apply to 
more than one of the figures. 

Accordingly, if we add to the general rules for the 
syllogism the special rules, we can formulate them as 
follows : — 



126 ELEMENTARY LOGIC 

General Rules. — 

1. One premise at least must be universal. 

2. One premise at least must be affirmative. 

3. If one premise is particular, the conclusion must 
be particular. 

4. If one premise is negative, the conclusion must 
be negative. 

Special Rules. — 

1. In Figs. I, III, and IV the minor premise 
must be affirmative, and in Figs. I and IV the major 
premise must be universal. 

2. In Fig. II one premise must be negative. 

3. In Fig. Ill the conclusion must be particular. 
There remains one form of the irregular syllogism 

for which special rules are required. It is the sorites. 
Examination of the Aristotelian sorites shows that the 
chain is broken, if any premise excepting the last is 
negative; also, that no premise excepting the first can 
be particular. Hence, for this sorites two rules are to 
be observed: i. Only one premise, and that the last, 
can be negative. 2. Only one premise, and that the 
first, can be particular. Inspection of the Goclenian 
sorites makes it evident that it is only the first premise 
that can be negative, and the last premise only can be 
particular. Accordingly, the two rules for this sorites 
are: i. Only one premise, and that the first, can be 
negative. 2. Only one premise, and that the last, can 
be particular. 



CHAPTER IX 

FALLACIES IN DEDUCTIVE REASONING 

Section 24 
description of fallacies 

A FALLACY is an error in reasoning or inference. It 
consists in the violation of some principle or condition 
of valid inference. 

There are two principal sources of fallacies : — 
(i) Misapprehension of the terms of the premises, or 

of the evidence supplied by the premises. 
(2) A misapprehension of the principles and conditions 

of right inference. 
Fallacies are therefore divided into two main classes, 
according to the source and nature of error committed. 
Material and Formal fallacies. In the material fal- 
lacies the error lies in the subject-matter, in some 
confusion of meaning in special terms or in a misap- 
prehension of the meaning and evidential force of the 
premises. Hence, the name material to designate this 
class of fallacies. In formal fallacies, the error is com- 
mitted in the process of inference, in proceeding from 
the premises to the conclusion. Accordingly, we can 

127 



128 ELEMENTARY LOGIC 

say, material fallacies give us the wrong premises from 
which the inference proceeds; formal fallacies draw 
the wrong conclusions from the given premises. Two 
examples will make clearer this distinction. Here is 
a fallacious argument : " A college graduate is sure of 
the appointment ; I am a college graduate ; and there- 
fore I am sure of the appointment." 

Now, the error in this reasoning Hes in overlooking 
a difference in the meaning of the middle term, a college 
graduate. 

In the first premise, it is a college graduate who has 
those particular qualifications or attributes which will 
insure his obtaining the appointment. In the second 
premise, a college graduate is a man who need have 
only the attributes which every other college graduate 
has, and in consequence of which I am a college 
graduate. Therefore, that which makes me a college 
graduate does not necessarily make me the particular 
species of college graduate that will secure this appoint- 
ment. 

The next example gives, we shall see, a different 
kind of fallacy. " Those who think this man is innocent, 
think he should not be punished ; you think he should 
not be punished; therefore, you think he is innocent." 
In this reasoning the fallacy lies somewhere on the way 
between the premises and the conclusion; and a little 
inspection will find it. The argument is a syllogism 
in Fig. II, and both its premises are affirmative; and 
this violates the special rule which requires that, in 
this figure, one premise shall be negative. This fal- 



FALLACIES IN DEDUCTIVE REASONING 1 29 

lacy is therefore formal, while the preceding fallacy is 
material. 

I. Material Fallacies. — Having defined fallacies, 
and explained the chief distinction between them, I 
shall now describe the fallacies of the first class, the 
material fallacies. These fallacies, we have seen, con- 
sist either in a wrong interpretation of the terms, or of 
the propositions employed. The description of these 
fallacies will be more easily followed if we carefully 
examine typical specimens of them. As the first case, 
(i) take the following: "All presuming persons are 
contemptible; this man is therefore contemptible, 
because he presumes that his opinion is correct." 

The error in this argument has its root in a double 
meaning of the term presume; in the first premise it 
has a meaning to which is attached contemptibleness ; 
in the second premise it does not have that meaning. 
The fallacy consists in overlooking this difference of 
meaning, or in assuming an identity of meaning where 
it is not. Giving to this term its proper meaning in 
each proposition, it is evident that these two propo- 
sitions are not so connected that any conclusion can 
be drawn from them ; in other words, they are not real 
premises, but two propositions which have nothing 
to do with each other. 

(2) The next case for examination is the following: 
"Pine wood is good lumber; matches are pine wood; 
therefore, matches are good lumber." In this argu- 
ment there is no ambiguity in the middle term, as was 
the case in the preceding argument; and yet there is 



I30 ELEMENTARY LOGIC 

a confusion of distinct things. If we examine the 
middle term in each premise, we shall discover the 
nature of this confusion. The assertion in the first 
premise is, pine wood possesses those properties or 
attributes which are identical with the properties essen- 
tial to good lumber; the assertion in the second prem- 
ise is, that matches possess those same attributes or 
marks which constitute that kind of substance called 
pine wood. Now this argument assumes that the attri- 
butes which make matches pine wood are identical 
with the attributes which make pine wood good lum- 
ber; or to express it in technical terms, the argument 
confounds generic marks with marks which are not 
generic, and which, not being essential to the conno- 
tation of the name pine wood, are accidents. The 
marks which make matches pine wood are generic 
marks; the marks by virtue of which pine wood is 
good lumber are not its generic marks, and therefore 
not the marks it has in common with matches; these 
marks are accidental to its connotation. Therefore 
the two propositions on which this argument is based 
establish no connection between pine wood and good 
lumber. That which makes matches pine wood does 
not necessarily make it good-pine-wood-lumber. 

(3) The next case to be examined admits of two 
explanations; it is the following: "The holder of a 
ticket in a lottery is certain to draw the prize ; and, since 
I hold a ticket, I am certain of drawing the prize." One 
way of explaining this fallacy is to make it a case of 
simple ambiguity of the middle term, a ticket; the term 



FALLACIES IN DEDUCTIVE REASONING 131 

meaning in the first premise a certain ticket, and in 
the second premise it means any one ticket. The other 
interpretation, and, in my judgment, the correct one, 
makes the fallacy one of the same species as the one 
just described, a fallacy consisting in the confusion 
of essential with accidental marks. The argument 
wrongly assumes that the marks which make me a 
ticket holder make me a prize-drawing-ticket-holder, 
which is not necessarily the case. 

(4) The fallacy in the next argument, though one of 
confusion, is of a slightly different sort from those 
already described. "The Germans are beer drinkers; 
and since Hans is a German, I infer he is a beer drinker." 
The technical name of this fallacy is, arguing from a 
general rule to a special case; and this term well 
describes the fallacy. The particular source of this 
fallacious inference is a confusion of a general statement 
with a universal statement. The universal permits 
no exceptions ; the general allows a considerable num- 
ber of exceptions. Some one might maintain that the 
fallacy in this argument belongs to the class of formal 
fallacies ; that it is a case of a syllogism in Fig. I having 
a particular major premise. The first proposition means 
some Germans are beer drinkers; hence the syllogism 
becomes, " Some Germans are beer drinkers : Hans is 
a German; therefore Hans is a beer drinker," a con- 
clusion which, according to the rule for syllogisms in 
Fig. I, is inadmissible. It is better, however, to regard 
this argument as a case of material fallacy, since the 
error is primarily one of misinterpretation. 



132 ELEMENTARY LOGIC 

(5) Another variety of confusion is presented in the 
following argument: "All the trees in the park make 
a dense shade; that pine tree is a tree in the park; 
therefore it makes a dense shade." The error in this 
argument consists in overlooking the difference between 
the trees being all taken together, as they are in the 
first premise, and the trees considered individually, 
or distributively, as they are so considered in the second 
premise. The technical name of this fallacy is the 
fallacy of composition. 

Were I to argue that since no one of the trees in the 
park makes a dense shade, there is no dense shade in 
the park, I should commit the converse fallacy, that 
of division, which consists in overlooking the difference 
between considering the individuals of a class separately 
and these individuals taken together. 

(6) The next case to be examined presents a mate- 
rial fallacy, but a fallacy of quite a different character 
from those we have examined. Suppose the following 
argument is presented: "It must be on the whole a 
good thing to allow every man an unbounded freedom 
of speech, because it is highly advantageous to the 
community that each individual should enjoy a liberty, 
perfectly unlimited, of expressing his sentiments." 

The fallacy in this argument does not consist in a 
confusion of things that are different, but in the failure 
to perceive an identity where it exists, in this case the 
identity in meaning between the conclusion and a 
premise, or that which is to establish the conclusion; 
in other words, the argument begs the question; it 



FALLACIES IN DEDUCTIVE REASONING 1 33 

moves in a circle. The source of this error is a mis- 
conception of that which is needed in order to establish 
the given conclusion; hence, something is offered as 
proof which is the thing to be proved. 

Let us take another example of this kind of fallacy. 
"That doctrine should be condemned because it is 
heresy." One who does not accept this argument 
could not accuse the reasoner of begging the question 
in the same sense in which the preceding argument 
is called begging the question. Certainly, this reason- 
ing is not open to the criticism that it moves in a circle ; 
the only reply to this argument is that it makes use of 
an unwarranted premise, that it assumes without right 
the truth of the proposition, " All heresy should be con- 
demned." Hence, begging the question, or petitio 
pHncipii, as it is designated, is done in two ways: 
either by taking as proof of the conclusion that which 
is virtually the conclusion itself; or, by taking for a 
premise a proposition the truth of which needs first to 
be proved. 

Another technical designation for the first species of 
petitio principii is circulus in probanda. Of these two 
forms which this fallacy assumes, it is not difficult to 
recognize and estabhsh the first ; but it is by no means 
the same with the second form. When is the fallacy 
of unwarranted assumption committed? I think no 
definite or absolute answer can be made to this ques- 
tion. A reasoner is always liable to the charge of 
making unwarranted assumptions. There are, how- 
ever, three ways by which one can safeguard his rea- 



134 ELEMENTARY LOGIC 

soning from this attack : the reasoner must confine his 
premises to self-evident propositions ; or he must be at 
pains to estabhsh each proposition that he is to use 
for premises; or he must get the acceptance of the 
premises at the outset. 

(7) The last case of fallacious reasoning illustrates 
another species of the fallacy due to misapprehending 
the premises. " The more correct the logic, the more 
certainly will the conclusion be wrong, if the premises 
are false; therefore where the premises are wholly 
uncertain, the best logician is the least safe guide." 

In this argument the proposition, "The best logician 
is the least safe guide," is substituted for or mistaken 
for the proposition, "the best logician will draw no 
certain conclusion where the premises are uncertain." 
This last proposition is the only admissible conclusion 
from these premises. The fallacy consists in assuming 
that two propositions are identical, which are really 
different. This fallacy, technically called ignoratio 
elenchi, is closely allied to that of petitio principii; 
however, this distinction can be made between them: 
in petitio principii there is ignorance of that which 
constitutes proof of a given proposition ; in ignoratio 
elenchi there is ignorance of that which is to be proved. 
In petitio principii wrong propositions are used to 
establish the conclusion; in ignoratio elenchi some 
other proposition is taken for the conclusion. 

The following are other designations of this fallacy : 
arguing to the wrong point, irrelevant conclusion, argu- 
mentum ad hominem, argumentum ad populum, etc. 



FALLACIES IN DEDUCTIVE REASONING 135 

The two last expressions designate varieties of this 
fallacy; one being some retort, or attack upon the 
reasoner, instead of an answer to his argument; the 
other being the substitution of appeal to the prejudices 
or passions for a refutation of the argument presented. 

Ignoratio elenchi may occur in serious reasoning; 
but it is more commonly, however, the expedient of 
the hard-pressed opponent, the disingenuous advocate, 
or pubhc debater who, instead of meeting an argument, 
seek to divert the minds of their hearers from the issue. 
The reply of a barrister to a request from his attorney 
in a certain case illustrates the character of this fallacy : 
"We have no case; abuse the plaintiff's attorney." 
As a further illustration of the ad hominem variety of 
this fallacy, we have the following: A member of the 
legislature is advocating the passage of a certain bill; 
his opponent, instead of showing that the proposed 
measure is not desirable, offers as argument the incon- 
sistency of the member in now advocating a measure 
which only a short time ago he opposed. 

Again, suppose one is arguing that a certain measure 
should become a law, and the reply is, "This is a bad 
measure, for it is supported by bad men; see what 
sort of men are in favor of this law; you will be in fine 
company in supporting such a measure!" Here, we 
have an example of the ad populwn variety of the 
ignoratio eclenchi. 

2. Formal Fallacies. — These fallacies consist, as 
we have seen, in a violation of the rules for valid infer- 
ence. Some of these fallacies occur in the simple form 



136 ELEMENTARY LOGIC 

of deductive reasoning called immediate inference; 
they arise from misapprehending the principles of 
obversion, conversion, contraposition, added deter- 
minants, and the relations of opposition. Thus, it 
is not uncommon for the student to confound ob- 
version with a proposition that resembles the obverted 
one, but is wholly different in character. 

If, for the obverse of the proposition, "All A is 5," one 
gives, "What is not A is not 5," he commits the fallacy 
of wrong obversion, this proposition being by no 
means the same as, " No A is not jB." 

So with other forms of immediate inference ; each is 
exposed to a fallacy. The following proposition was 
once given to a class in an EngHsh university, "A 
stitch in time saves nine," and of a large class but few 
gave the right converse, most merely giving the proposi- 
tion with the verb in the passive voice; Thus, "Nine 
stitches are saved by a stitch in time." 

Again, the A proposition is not infrequently converted 
without limiting the extension of the new subject term. 
It is by no means needless to caution students against 
erroneous inferences based upon the various relations 
of opposition. The contrary relation is a snare to more 
than a few incautious reasoners ; more than half the 
students in a class in logic have inferred the truth of 
one contrary from the untruth of the other contrary 
proposition, and to quite as large a proportion of the 
class it seemed correct to assert that both subcontraries 
can be false. 

Coming now to the fallacies in mediate or syllogistic 



FALLACIES IN DEDUCTIVE REASONING 1 37 

inference, we shall see that they all consist in a violation 
of the conditions of valid syllogisms. We will first note 
those which occur in hypothetical syllogisms. The only 
fallacies to which this form of reasoning is liable are 
the fallacy of affirming a consequent and that of deny- 
ing a condition or antecedent. 

It is the categorical syllogism that furnishes most of 
the formal fallacies, and these we will now examine. We 
shall find that these fallacies consist either in a non- 
inclusion of a term where such inclusion is necessary to 
the inference, or in the failure to perceive that a term 
is not taken in its full extension, or in the failure to ob- 
serve the quantity and quahty of the premises. Accord- 
ingly, formal fallacies in the categorical syllogisms fall 
into these classes : — 

(i) fallacies of nonsubsumption, 

(2) fallacies of undistributed middle term, 

(3) fallacies of wrong quantity and quahty. 
Nonsubsumption may affect either the middle or 

the major term. A syllogism in Fig. I with a nega- 
tive minor affords an example of nonsubsumption in 
the middle term; thus, "All A is B; no C is ^ ; 
therefore, no C is D, is a false syllogism, and the 
fallacy in it is technically the fallacy of nonsubsump- 
tion in the minor term. Nonsubsumption in the 
middle term occurs in syllogisms with two negative 
premises. 

As an example of the fallacy of undistributed middle 
term take a syllogism in Fig. II, with two affirmative 
premises : — 



138 ELEMENTARY LOGIC 

AU^ is 5; 

AUCisB; 

Therefore all C is yl. 

The fallacy here consists in not observing that the 
middle term B is not taken in its full extension. 

The fallacies of the third class have no technical 
designations. Cases of them are the following: — 

(i) whenever a universal conclusion is drawn from 
premises in which one proposition is particular, 

(2) when a particular conclusion is drawn where a 
universal is possible, 

(3) when an affirmative conclusion is drawn from 
premises containing one negative proposition. 

There are no accepted technical designations of the 
fallacies in this last group. It would describe them 
well did we call those under — 

(i) fallacies of proving too much; those under 

(2) fallacies of proving too little; and those under 

(3) fallacies of a proposition of the wrong quahty. 

Section 25 

classification and technical designation of 
fallacies 

The various kinds of fallacies in deductive inference 
have been described and incidentally to most of them 
have been given their technical designations. To 
some extent a classification has also been given. I shall 
not follow the usual practice, and attempt to give a com- 



FALLACIES IN DEDUCTIVE REASONING 1 39 

plete classification. It is not, I think, possible to do 
so, for the reason that some of the fallacies, especially 
some of the material fallacies, cannot be successfully 
classed. They do not belong decisively and indisput- 
ably to any one of the classes into which they are put 
by this or that logician. 

Nor is the matter of correct designation so simple as 
it would seem. However, it may be advantageous to 
the student to have the suggestion of a plan or scheme 
of classification which he can carry out, or reconstruct 
in the interest of a more satisfactory classification. 

I will outhne such a plan, and then add some obser- 
vations upon the principles of the classification I suggest 
and also upon the technical designations of certain 
fallacies. 

The fallacies of deductive inference are of two kinds ; 
they fall into two main groups, one group comprising 
the formal fallacies, the other group comprising the 
material fallacies. The formal fallacies again subdi- 
vide into fallacies of immediate inference and fallacies 
of mediate or syllogistic inference. The fallacies of 
immediate inference can be further separated into falla- 
cies of equipollence and fallacies of opposition. The 
fallacies of equipollence include the following : — 

1. False ob version. 

2. False conversion. 

3. False contraposition. 

4. False added determinants. 

The fallacies of opposition comprise the follow- 
ing:— 



140 ELEMENTARY LOGIC 

1. Fallacy contrary. 

2. Fallacy contradictory. 

3. Fallacy of subcontrary. 

4. Fallacy of the subalterns. 

The fallacies of mediate inference fall into the follow- 
ing subdivisions : — 

1. Fallacy of nonsubsumption. 

2. Fallacy of nondistribution of middle term. 

3. Fallacy of wrong quahty and quantity in the con- 

clusion. 

4. Fallacy of wrong minor premise in hypothetical 

syllogism. 
The material fallacies fall most conveniently into the 
following subdivisions : — • 

1. Fallacy of confusion. 

2. Fallacy of mistaken proof or evidence. 

The confusion fallacies again subdivide into : — 

1. Simple ambiguity. 

2. Composition and division. 

3. Accident. 

The fallacy of mistaken proof subdivides into : — 

1. The fallacy of petitio principii or begging the 

question. 

2. Fallacy of ignoratio elenchi or irrelevant conclu- 

sion. 

The fallacy of petitio principii presents two varieties, 
unwarranted assumption and circle in proof, circulus 
in probanda. 

It is hardly worth while to specify the varieties of the 
ignoratio elenchi fallacy. 



FALLACIES IN DEDUCTIVE REASONING 14I 

I add a few words in explanation of some things in 
this scheme of classification. Accident fallacies are 
those which give most trouble, and those about which 
there is disagreement among logicians. The term 
should be applied to those fallacies and to those only in 
which the error consists (as I have shown in the analysis 
of these fallacies), in the confusion of generic or essen- 
tial marks with accident, or nonessential marks. This 
fallacy does not consist, as Jevons appears to think, in 
arguing from a general rule to a special case, and the 
converse, arguing from a special case to a general rule. 
These two fallacies are species of the accident fallacy, 
the technical names for them being a dido simpliciter 
ad dictum secundum quid and a dicto secundum quid ad 
dictum simpliciter. 

The fallacy of composition and division is by some 
logicians regarded as a species of accident fallacy, but 
incorrectly. The source of this fallacy is the confusion 
of a class composed of individuals that are taken to- 
gether with a class the members of which are taken dis- 
tributively. The fallacy is based upon the assumption 
that one can always predicate the same thing of the 
individuals of a class taken separately that one can 
predicate of them taken together, and conversely, one 
can predicate of things taken together what one can 
predicate of them taken separately. 

The exact designation of the fallacy which includes 
petitio principii and ignoratio elenchi is not easy to 
find. I think, however, it is more accurately described 
by the terms I have used than by those in common 



142 ELEMENTARY LOGIC 

use. The essence of this fallacy is misconception of 
what the premises prove, and what sort of premises are 
needed to prove the thing we seek to establish. Hence, 
this misconception takes two forms, — either a miscon- 
ception of what is requisite for the proof sought, or a 
misconception of what is to be proved, and therefore 
of what the given premises prove. In the former case 
we have petitio principii, in the latter, we have igno- 
ratio elenchi. 

Section 26 

the value of the syllogism 

Regarding deductive reasoning, and the syllogism in 
particular, opposite views are held. Some maintain 
that the syllogism is a useless survival of mediaeval logic ; 
that it is no legitimate form of inference, being a mere 
petitio principii. Others have maintained that the 
syllogism is indispensable to valid reasoning, being the 
only form in which inference can be expressed if it is 
to be clearly valid. The truth lies between these ex- 
treme views. The syllogism has important uses, and 
serves purposes of no inconsiderable value in the dis- 
cipline of thought. But it is true, at the same time, that 
this form of inference has rather narrow limits. It is 
not applicable to all cases of genuine inference. Some 
of these cases cannot without awkward and unnatural 
constructions be made to take the syllogistic form; 
other cases either do not come under the forms of the 
syllogism, or plainly violate its canons. 



FALLACIES IN DEDUCTIVE REASONING 1 43 

In discussing the function and value of the syllogism, 
I shall first consider its hmitations and its defects ; and 
secondly, I shall show in what consists the value of this 
much-decried instrument in reasoning. 

The syllogism has a hmited range of utihty. It is 
useless and embarrassing in some cases of deductive 
reasoning. Such are inferences based upon relations of 
space, time, and quantity. Here are some instances: 
given the position of A at the right of B, and the posi- 
tion of C at the right of A, with this datum we at once 
draw the inference that C is at the right of B. Now, 
let us put this reasoning into the syllogistic form, and 
we shall get some such construction as the following: 
" What is at the right of A is also at the right of that posi- 
tion in space of which A is at the right, which in this 
case is 5 ; C is at the right of A ; therefore C is also at 
the right of 5." 

Such a syllogism is certainly a clumsy and unnatural 
construction, and quite needless, since the property of 
space which is the foundation of all such inferences is 
directly perceived, and once perceived such a round- 
about way of reaching a conclusion is useless. 

Take as another instance, " A follows B in time, C 
follows A." These premises give at once the conclu- 
sion, C follows B, the foundation of inferences of this 
sort being clearly the law of time sequence. Now, ex- 
press this inference in the traditional syllogism and this 
rather awkward structure is the result, " Whatever follows 
A in time, follows also that which A follows, which in 
this case would he B; C follows A ; therefore C follows 



144 ELEMENTARY LOGIC 

5." One more instance: ''A is greater than B; C 
is greater than A ; therefore C is greater than 5." 

A glance at these propositions, while it assures us that 
the third is the conclusion from something contained 
in the two preceding propositions, shows us that these 
propositions do not constitute the syllogism of formal 
logic, for there is no major premise, or rather the major 
premise is implied and not expressed. Now, supplying 
this premise, we get the following: "Whatever is greater' 
than a given quantity, say B, is greater than that quan- 
tity than which this quantity is greater; C is greater 
than this given quantity A ; therefore C is greater than 
B." It can hardly be maintained that such a form of 
reasoning facihtates the passage of thought from datum 
to conclusion. The direct perception of the relation of 
quantity is the bridge over which thought passes easily 
and surely to the conclusion in such cases. 

But not only is the syllogism useless and a hindrance 
in certain cases of reasoning, there are cases of genuine 
inference which the syllogism does not recognize, which 
fall outside of its forms of admissible inference. Here 
are a few instances of such inferences : 

(i) "Horses are vertebrates; dogs are vertebrates." 
As the student at once perceives, these are the prem- 
ises of a syllogism in Fig. II, and no conclusion is 
admissible according to the rule for valid syllogisms in 
this figure. From these premises nothing can be deter- 
mined respecting the relation between horses and dogs. 
But is it after all true that this datum yields no signifi- 
cant and rational belief? 



FALLACIES IN DEDUCTIVE REASONING 145 

Let US suppose that horses and dogs are names for 
a large number of individuals which, on the basis of 
common properties, have been, for convenience, put into 
these two classes. Let me further suppose I have ob- 
served that all the individuals in these two classes have 
a number of properties in common with a third class of 
individuals called vertebrates, or that both these classes 
can be put into this larger group; now, does not the 
fact that dogs and horses possess a number of properties 
in common justify me in beheving in some degree that 
they have other properties in common; or, in other 
words, that they agree in other respects, so that the one 
class may be included in the other ? 

This is the sort of reasoning on which we proceed in 
matters of practical interest, and which has a recog- 
nized value in science. I conclude from the datum, in 
this case, that horses and dogs probably agree in other 
attributes, in addition to those which make them both 
vertebrates. My reasoning is to this effect : if these two 
classes of animals agree in these respects, they probably 
agree in other respects. 

This conclusion is a probabihty; but the syllogism 
does not recognize mere probabilities. But probabihty 
is a rational belief, and the inference that leads to 
such a proposition is not less logical in character 
than is the inference which the syllogism sanctions. 
Probability is the guide of hfe, and what guides Hfe 
must be rational. 

(2) One more illustration of a genuine inference, 
which is, according to the syllogistic canons, illegitimate. 



146 ELEMENTARY LOGIC 

I observe that a number of particular substances, A, 
B, C, D, etc., exhibit a certain mode of behavior, say a 
chemical reaction of some sort, and I draw the inference 
from these particular facts that all substances Uke these 
will exhibit the same reaction. Now, such an infer- 
ence is forbidden by the rules of the syllogism, or rather, 
the syllogism does not recognize such an inference. 
From such particular premises no conclusion can be 
drawn according to the conditions of a vallid syllogism. 
But here again my inductive inference, giving a prob- 
able conclusion, is as logical a process as the processes 
which are valid according to the syllogism. This gen- 
eralization from experience, like the inference from 
analogy, is the expedient of daily life, and as rational 
a way of dealing with the matters of our experience as 
any of the methods of formal inference. Thus is it 
shown that there is a large class of genuine and useful 
inferences which he outside the field of the syllogism. 

But not only is it true that many cases of vaHd in- 
ferences are not recognized by the syllogism, there are 
also cases of valid inference which it is claimed by 
some logicians violate the syllogism. Paradoxical as it 
sounds, a vahd inference is possible from two negative 
premises. Take as an example of such a syllogism the 
following : — 

X is not a knave ; 
X is not a fool ; 

therefore, he who is not a knave is not necessarily a 
fool ; or knavery and folly do not in all cases go together ; 



FALLACIES IN DEDUCTIVE REASONING 1 47 

or because a man is not a knave, do not conclude he is 
a fool. Now, here is a syllogism in Fig. Ill with two 
negative premises, which, according to the rules of the 
syllogism, cannot give a conclusion. I will leave it to 
the student's judgment to decide whether there is or is 
not a conclusion from these premises. 

Let us change these premises so as to get a valid 
syllogism: — 

X is a knave ; 
X is a fool ; 

therefore, some knaves are some fools, or knavery and 
folly sometimes go together; and where you find one 
you may find the other. Let the student compare 
these two conclusions, and say which of them is the 
more significant or suggestive, the one from the affirma- 
tive, or the one from the negative, premises. 

Let us add one more case which brings out more dis- 
tinctly the legitimacy and the value of conclusions from 
negative premises. "None of the men in the town A are 
rich ; none of these men are unhappy ; therefore, riches 
are not necessary to happiness; people can be happy 
who are not rich. Do not, therefore, conclude that a 
man who is not rich is not happy." 

Here is a conclusion from negative premises that is 
not only admissible, but quite as important as would 
be the conclusion were one or both of these premises 
afiirmative. Defenders of the traditional syllogism 
have maintained that these alleged cases of valid con- 
clusions from two negative premises are not really 



148 ELEMENTARY LOGIC 

violations of the syllogism ; because there are not, as 
is alleged, two negations in the premises, one at least 
of the premises, it is held, is reaUy affirmative. 

This defence is vaHd against some of the instances 
which have been given by critics of the syllogism; but 
it does not avail against such cases as the first given 
above. The premises there are as genuine negations, 
as can be found, and a significant conclusion follows. 
A better defence of the vahdity of the rule which forbids 
a conclusion from negative premises is to observe the 
limits within which it is valid, the exact nature of the 
relation between things which the syllogism contem- 
plates. 

Turning again to our first example, let us inter- 
pret the two propositions according to the relation 
on which the traditional syllogism is based. Now, 
there is one sense in which these premises do not per- 
mit a conclusion, or rather there is one sort of conclusion 
that does not certainly follow from these premises. We 
cannot from these two negative propositions reach a 
definite conclusion respecting the relation between these 
two classes, fools and knaves; we cannot determine 
whether some knaves are or are not in the class of fools. 
It is just this relation between two classes that the cate- 
gorical syllogism contemplates, and consequently in such 
a case as the one given no certain conclusion follows. 
Another fact must be kept in mind, the only conclusions 
that the syllogism recognizes are those that are certain ; 
the syllogism knows nothing of probabihty. When 
these two facts are borne in mind, the contradiction of 



FALLACIES IN DEDUCTIVE REASONING 1 49 

the syllogism which some logicians see in such cases is 
apparent rather than real. These cases of conclusions 
from negative premises, Uke the cases we have noted 
above, do not fall within the field of syllogistic infer- 
ence, as that inference is defined in formal logic. 

So much for the limitations of the syllogism. Now 
let us turn to the positive side, to the uses of the syllo- 
gism. 

There is one thing which in all reasoning is of primary 
importance, — definite premises. There must not be 
vagueness and obscurity here, if there is to be any clear 
and definite issue from the premises. One value 
of the syllogism is that it enables and compels the rea- 
soner to make the first step in argumentation definite 
and exact. The syllogism is an instrument by the aid 
of which the exact meaning and scope of the proposi- 
tions which form the datum can be determined. For this 
purpose, no better instrument has been devised than the 
syllogism. The syllogistic analysis and coupling of 
propositions is the most effective means yet devised by 
which the premises are defined and made perfectly 
clear. 

Again, in the process of inference, the syllogism is 
the most serviceable instrument for keeping the way 
clear from premises to conclusion. The principles and 
rules of syllogistic inference are guideboards which read 
so plainly that only the heedless or very stupid reasoner 
can miss his way, so thoroughly is he safeguarded 
against misleading ways. No more simple or efficient 
instrument has been discovered for detecting errors 



150 ELEMENTARY LOGIC 

into which our own thinking may fall, and errors in 
which the sophistry of another reasoner may try to 
entangle us. 

This discipline of thought which the use of the syllo- 
gism yields has been too Hghtly appreciated. The 
abihty to go at once and unerringly from a given propo- 
sition to all that is implied in that proposition, and from 
two propositions to all that follows from their admission, 
is no small or easy acquisition ; and it is an abundant 
.justification of the syllogism and a sufficient reason for 
its retention in the training of the intellect that it gives 
this ability, without which one is not a good reasoner. 



PART TWO 

THE LOGIC OF SCIENCE 
CHAPTER X 

INTRODUCTORY 

The division of our study in logic is based upon the 
twofold aim in thinking, consistency and truth of fact. 
The customary title Inductive Logic is not, it seems to 
me, a fortunate one ; first, because it implies that there 
are two kinds of logic, each with principles of its own ; 
whereas logic and logical principles are of one and the 
same nature, whatever may be the subject-matters to 
which they are applied. The fundamental purpose of 
logic is to ascertain and apply the principles which are 
regulative for our thinking. Now, whatever may be 
the special aim of this thinking, this main function of 
logic is the same. 

The title Inductive Logic is not fortunate for another 
reason: the term inductive has two meanings, and 
taken in one meaning, it is too narrow to define this 
division of logic ; taken in the other meaning, it requires 
an extension of the term which ought not to be given 

151 



152 ELEMENTARY LOGIC 

it. Induction means an inference which proceeds from 
particular facts of observation. Induction in the usage 
of some logicians also means those various processes 
by which science explains the facts of nature, and 
which are employed in all investigation. To make 
this term cover all these processes and methods by 
which scientific knowledge is attained is to extend the 
meaning of the term beyond its proper limits. The 
title I have given this division of logic marks distinctly 
the aim proposed, — an exposition of the principles of 
logical thinking which are employed in science. 

Section 27 
the meaning of science 

A successful execution of the task now undertaken 
requires that we have at the outset the right conception 
of science, — its subject-matter, its aim, and its limits. 
I shall briefly discuss these before proceeding to the 
exposition of the logic of science. 

I. The Facts of Physical Science. — The objects of 
scientific knowledge are phenomena and phenomena 
only. Phenomena are things which are perceived or 
which can under supposable conditions be perceived; 
they are events which occur; they are changes which 
take place, or processes, such as motion, which go on 
and can be observed and be matters of exact measure- 
ment and description. The world from the point of 
view of science and for the aim of science is a phenome- 
nal world. 



INTRODUCTORY 1 53 

2. The Province of Scientific Explanation. — Scien- 
tific explanation consists in finding for these facts and 
events the most general laws to which they conform. 
A phenomenon is scientifically explained when it is 
shown to be an instance of a general law, apphcable 
to all phenomena of like description ; or when this phe- 
nomenon is referred to some definite antecedent condi- 
tion, which, being given, this phenomenon invariably 
follows. 

For the right understanding of scientific explanation, 
two terms must be accurately defined. One is the term 
cause, the other the term law. For the purpose of 
science, a cause need only be an antecedent phenome- 
non on which a given phenomenon invariably depends. 
Invariable antecedence in time is the only necessary 
mark of causal connection. The term law in science 
means a uniform and invariable order in which phe- 
nomena occur. 

Laws of nature are statements of the uniformities of 
succession and existence among phenomena, and the 
ideal of science is the reduction of these uniformities to 
the fewest in number and the simplest in character. 
Laws are, therefore, not things which exist or have any 
meaning apart from phenomena; they are only de- 
scriptive formulae by the aid of which science describes 
in the simplest and most comprehensive terms the 
manner in which the phenomena of the world occur. 
Laws do not prescribe how things shall take place; 
they are formulae for describing how things do take 
place. 



154 ELEMENTARY LOGIC 

3. The Limits of Scientific Explanation. — Science is 
limited in two respects : first, in respect to the subject- 
matter of its explanation; and secondly, in respect to 
the explanation it gives. There are some things which 
science presupposes as the necessary condition of its 
explanations. It presupposes, for instance, the uni- 
formity of nature. This is the working postulate of 
science; without it not a step can be taken. But this 
principle on which science depends is not something 
which science has discovered ; it is the mind's trust in the 
rational character of the world and the adaptation of the 
world to our purposes and needs; it is an essentially 
ethical faith that Nature will not disappoint our 
expectations, nor put us to intellectual confusion in 
our attempts to know and practically to control our 
world. 

Science presupposes such things as matter, force, 
space, time, etc. The exact meaning of these concep- 
tions Hes outside the field of science. It is the function 
of science to describe in the simplest and fewest pos- 
sible terms the motions of that which we call matter; 
but science does not undertake to say what matter is. 
Science explains the phenomena of fife, the evolution 
of living beings; it describes their various behaviors; 
but it does not tell us what life is, whence it comes 
or whither it goes. Science describes the various func- 
tions of mind, mental phenomena; it formulates the 
laws in accordance with which they occur; it investi- 
gates the various connections between these phenom- 
ena and phenomena of the physical order. Science 



INTRODUCTORY 1 55 

traces the evolution of mind from its simplest dis- 
cernible manifestation to its highest and most complex 
functions ; but science leaves unanswered the question, 
what is the mind; what is the thinking, feeling, and 
willing being called ego, mind, soul, self. 

Nor is science a final or complete explanation; it 
stops short of the goal of rational explanation. There 
are two questions which man as a thinking being neces- 
sarily asks about everything, — the question of whence 
and how, and the question of why, what for. It lies 
within the province of science to answer all questions 
of genesis, all questions of how. It does not lie within 
her province to answer the other more significant and 
often more urgent questions. 

The function of science, we have seen, is description. 
So far as the world is a describable world it belongs to 
science. But there is more than a world of description. 
There is also a world of valuation ; there are meanings 
and values of which science can take no account. At 
this boundary line between the answer to the questions 
whence and how, and the questions why and what for, 
science submits to the dictate, Thus far and no farther 
canst thou come. 

4. The Special Problems of Science. — Having ex- 
plained the nature of science and delimited its field, I 
will next explain the special problems that belong to 
science, and in a general way explain the methods by 
which these problems are solved. The first of these 
problems is the ascertainment of causal connection 
between known phenomena ; this connection is between 



156 ELEMENTARY LOGIC 

phenomena that are observed or can be made observa- 
ble by experiment. 

This problem belongs to the first stage of scientific 
explanation; and it arises out of the character of our 
experience, the way in which the world is directly given 
to our minds in simple sense perception ; and this prob- 
lem means the reconstruction of this rather chaotic 
world of direct experience, so far as to reduce its events 
and phenomena to some degree of uniformity of occur- 
rence. The search for causal connection is the attempt 
of rational thought to get behind mere appearances to 
the real world. 

The world of our immediate or direct experience is 
very unlike the world which science constructs or dis- 
covers. Order, unity, causal connection do not lie upon 
the surface ; they are not immediately presented to our 
senses; they must be sought for and constructed out 
of the data which our sense perceptions supply. "The 
order of nature," says Mr. Mill, "as perceived at first 
glance presents at every instant a chaos followed by 
another chaos. We must decompose each chaos into 
the single facts ; we must learn to see in the chaotic an- 
tecedent a multitude of distinct antecedents. . . . The 
regularity which exists in nature is a web composed of 
distinct threads, and only to be understood by tracing 
each of these threads separately." 

These first threads which science traces out are those 
of causal connection in the observable parts of this web ; 
and what renders this problem difficult is the fact that 
the causal connections are, to quote a statement from 



INTRODUCTORY 1 57 

another writer, "embedded in a mass of extraneous and 
irrelevant material from which it is our business to 
dissect them out." 

The second special problem of science is explanation 
by hypotheses. 

Science cannot stop with the first stage of explanation, 
if, indeed, that step can be called explanation at all, and 
not a mere preliminary step to explanation. All real 
explanation involves a step from the known to the un- 
known. Now this step is hypothesis, the essence of 
which is the supposition of the existence of something 
not seen, not yet known. An hypothesis is an ideal 
construction. By it thought goes beyond sense and 
conceives some reahty beyond the hmits of observation 
and experience. 

The justification of taking this step is that assump- 
tion which underhes all our knowledge of nature and 
all science, the continuity and orderhness of the world. 
Science assumes that the yet unobserved facts of the 
world are so related to known facts of experience, are 
so continuous with these facts present in experience, 
that both admit of ultimate description in terms of a 
common formula. And hence an hypothesis is an in- 
strument, a device, for this more comprehensive and 
accurate description of the phenomena and processes 
of the world. 

The great hypothesis of Newton was thus an ideal 
construction by means of which, not only the motions 
of particular bodies, but of every body in space, can 
be described, and those motions predicted for any 



158 ELEMENTARY LOGIC 

future time. So with the ether hypothesis; it is a 
grand fiction of the scientific imagination which is 
most serviceable in reducing to fewer and simpler pro- 
cesses a great number and variety of physical processes. 

It must be remembered, however, that hypothesis 
building is no work of fancy, but a task of serious 
thought. An hypothesis is no mere flight of the imagi- 
nation, but a venture of reason and at the bidding of 
rational thought. 

The third special problem of science is occasioned 
by those phenomena which, on account of their com- 
plexity and the obscure conditions on which they 
depend, do not admit of scientific explanation in the 
more exact meaning of that term; yet, because these 
phenomena do present certain uniformities in their 
occurrence, and admit to some extent of measurement 
and calculation, they come within the province of scien- 
tific method. 

These phenomena are of two sorts and, accordingly, 
two distinct methods are applied to them ; these meth- 
ods are technically known as calculation of chances, 
and the method of statistics. The former method is 
applied to those events which, taken singly, have no 
known cause, but which show a tendency to uniform- 
ity of occurrence and admit of calculation with vary- 
ing degrees of probabihty. The method of statistics 
is employed in deahng with those phenomena, which, 
taken in very considerable numbers or in masses, and 
observed for considerable periods of time, present cer- 
tain uniformities and certain persistent characteristics. 



INTRODUCTORY 1 59 

The calculations employed in this'' second method are 
not applied to the individuals which compose these 
aggregates, but to the aggregates only, considered in 
their mass character. 

Such are the special problems of science, and such 
in general the methods by which science effects or 
seeks to effect the solution of these problems. We pass 
now to the exposition of these problems and scientific 
methods in detail. 



CHAPTER XI 

THE ASCERTAINMENT OF CAUSAL CONNEC- 
TION BY OBSERVATION AND EXPERIMENT 

Section 28 
observation and experiment 

We have seen that order, unity, and causal connec- 
tion are not presented to our immediate experience; 
that Nature presents, instead, a web of tangled and in- 
terwoven threads, which present to direct perception a 
bewildering complexity. 

It is the first task of science to trace out these separate 
threads of causal connection, to disentangle them 
from the mass of connections which are not causal. 
Observation and experiment are the instruments we 
employ in the execution of this task. Accordingly 
these two operations must have our first consideration. 

I. Observation. — Experience teaches us that it is no 
easy thing to observe rightly and successfully, and yet 
observation is fundamental to all scientific knowledge. 
Carelessness, inaccuracy, or confusion here vitiate all 
the results that are gained by this first step, and that 
must furnish the data for the subsequent stages in 

160 



OBSERVATION AND EXPERIMENT l6l 

investigation. The difficulties of successful observation 
and the errors to which it is exposed are chiefly the 
following : — 

(i) The complexity of the phenomena themselves, 
the fact that every phenomenon we would observe and 
distinguish is embedded in a mass of coexisting phe- 
nomena ; it is just one fact in a very compHcated setting 
of incidents, a thread interwoven with countless other 
threads. Hence the difficulty of isolating the phe- 
nomenon we are trying to study, and the difficulty of 
eliminating the causal connection which this phenome- 
non sustains to some other phenomenon, from other 
concomitant conditions with which this phenomenon is 
not casually connected. 

(2) A second difficulty observation encounters arises 
from the limited time to which direct observation is 
confined. This time span, owing to the constitution 
of our mind, is very Hmited, and the difficulty this 
fact occasions is aggravated by the circumstance that 
in this brief time period a number of phenomena are 
occurring or existing simultaneously. 

(3) But successful observation is difficult for a third 
reason; successful observation depends upon the con- 
trol and persistence of attention ; and attention, unless 
disciphned by an energetic will, is easily distracted, and 
liable to be occupied with the nonessential concomi- 
tants of the phenomenon we are investigating. Add 
to the distracting influence of the multipHcity of simul- 
taneously occurring incidents, the influence of our pre- 
possessions, our subjective biases of various sorts, the 



1 62 ELEMENTARY LOGIC 

tendency to see what we are thus prepared to see, to 
observe what prior experience and habits dispose us to 
observe, — add these influences, and the difficulty of 
observation, arising from wrong attention, can be fully 
appreciated. 

It is owing to these difficulties I have pointed out 
that observation falls into various errors, the more 
common of which are the following : — 

(i) mal-observation ; this consists either in over- 
looking some important circumstance, or in a wrong 
perception of the circumstances in which a phenome- 
non occurs; 

(2) confusion of perception with something inferred 
from what is perceived. This form of error is exceed- 
ingly common, and one of the most subtle forms of 
wrong observation. How difficult it is to keep actual 
perception distinct from inference any one can appre- 
ciate who will introspect a little, or attend carefully to 
the relations of the same event by different people, 
equally well informed and equally conscientious observ- 
ers. One who is familiar with the proceedings of the 
courts is forced to confess that it is not easy for the 
most honest person to tell the whole truth and nothing 
but the truth. 

The Requisites of Good Observation. — To be a good 
observer three things are especially requisite: (i) 
accuracy and carefulness in perception, (2) power of 
sustained attention, (3) a good memory. This last 
requisite may seem to have nothing to do with obser- 
vation, which is confined to what is present ; but, so 



OBSERVATION AND EXPERIMENT 163 

narrow, so evanescent is our present perception that 
no one can observe a present fact and know in any 
degree what that fact is, who does not remember 
something at the same time. Part of every phenome- 
non we are trying to observe has shpped away into the 
past before we have really observed it; hence, our 
observation must have in it a constituent of memory. 
To hold completely and steadily in our grasp the 
immediate past is thus indispensable to an accurate 
and complete observation of anything that is occupy- 
ing our present thought. 

2. Experiment. — Observation alone, even were it 
ideally perfect, is inadequate to the task of analyzing 
the situations in which phenomena occur, and of ascer- 
taining in these situations what phenomena are causally 
connected. Hence, observation needs to be supple- 
mented by experiment, which is an artifice for enlarging 
and making obser\^ation more exact. Experiment is 
the instrument of science; it is an artificial treatment 
of phenomena, an intervention of our agency in the 
course of events, a subjecting of Nature to methods 
and tests of our own devising, in order to see more 
clearly what is the actual behavior of Nature her- 
self. 

I will now point out some ways in which experiment 
aids and supplements observation. 

(i) By making possible repeated instances of the 
same phenomena. Did we need to rely on observation 
alone we could learn httle of some phenomena, because 
they are of infrequent occurrence; but if we can by 



1 64 ELEMENTARY LOGIC 

experiment get a repetition of the same phenomenon, 
we are greatly aided in our observation of it. 

(2) By enabhng the observer to isolate the phenome- 
non under observation. We have seen that one dif- 
ficulty which observation encounters is the complexity 
of the conditions in which a particular phenomenon 
occurs. Experiment overcomes this difficulty by iso- 
lating this phenomenon; this it does either by eHmi- 
nating circumstances that are not causally connected 
with the phenomenon under investigation, or by pro- 
ducing different situations in which the same phenome- 
non occurs. 

(3) By the use of instruments. Here lies one of the 
great achievements of modern science, the employment 
of instruments for the measurement and calculation of 
the events and processes of nature. It is to this use of 
instruments that modern science largely owes her 
advance upon ancient science. It is mainly this 
employment of instruments that has made possible the 
accuracy and extension of observation on which this 
great advance of science depends. It is to the balance, 
the telescope, the microscope, the marvelous apparatus 
with which modern research is equipped, that we are 
chiefly indebted for the discoveries and expansion of 
knowledge which distinguish the past century from 
those which preceded it. 

Modern physical science is based upon the applica- 
tion of mathematics to the phenomena of nature; and 
instruments are methods of bringing these principles of 
mathematics into fruitful apphcation to nature. 



OBSERVATION AND EXPERIMENT 165 

We shall not be likely to overestimate these advan- 
tages of experiment over unaided observation in gaining 
a knowledge of our world, especially when we reflect 
upon this attitude to the universe which characterizes 
experiment, in contrast wuth the attitude that charac- 
terizes mere observation. In experiment man is no 
longer a passive observer, waiting for facts to be pre- 
sented to him; he actively intervenes in the course of 
events; he tries his universe, questions it, and pre- 
determines the sort of answers Nature will give to his 
questions, by selecting the questions he will ask. Man 
has found that it is the will of Nature that he that 
"asketh receiveth," he that "seeketh findeth, " to 
him that "knocketh it shall be opened." Man has 
learned by experiment that Nature is plastic to his 
action ; that she opens her mysteries to the importuni- 
ties of experiment ; that her word of assurance to him 
is " prove me and see if I will not reward thee." 

Section 29 

the regulative principles of observation and 
experiment. the so-called inductive methods 

The exact problem for observation and experiment 
is to ascertain which of the antecedents, or concomi- 
tants, of a given phenomenon is its cai^sal antecedent, 
or is one of its causal antecedents. 

The solution of this problem consists in the analysis 
of the situation in which the given phenomenon occurs ; 
and this analysis means separation between the con- 



1 66 ELEMENTARY LOGIC 

comitants of a phenomenon that are non -causal and 
the concomitants which are causal. 

The process is thus one of elimination, ehmination of 
non -causal circumstances from the totaUty of con- 
ditions in which the phenomenon under investigation 
occurs. Now, the inductive methods (as they are 
rather unfortunately named) are simply the ways in 
which this analysis and this elimination are effected; 
hence, it would be more appropriate to call them 
methods of analysis, or methods of elimination ; for that 
is precisely their function. 

These methods in their present formulation we owe to 
John Stuart Mill, although Mill did not discover them ; 
some of them were recognized by Bacon, and they were 
more fully recognized by Sir John Herschel. Nor 
must it be supposed that these m.ethods were invented, 
that the principles they formulate are a priori. Mr. 
Mill did not invent any of the canons or rules he for- 
mulated ; nor did he borrow them from other logicians ; 
he learned from the practice of men in different depart- 
ments of science, the methods they followed in their 
investigations and reasoning; and these methods of 
induction and their canons are only the formulation 
of the actual procedure and the accepted principles 
which men in science have always followed. 

I shall now present these methods substantially as 
they are formulated by Mill. 

I. The Method of Agreement. — This consists in 
observing the instances of the phenomenon under 
investigation, and noting in what single circum- 



OBSERVATION AND EXPERIMENT 167 

stance all these instances agree, while they differ 
in all the other material circumstances ; or, not- 
ing what single circumstance is always present, and 
the only one that is always present, when the given 
phenomenon occurs. Mill's canon for this method is, 
"If two or more instances of the phenomenon under 
investigation have only one circumstance in common, 
the circumstance in which alone all the instances agree 
is the cause or the effect of the given phenomenon." 

2. The Method of Difference. — According to this 
method there is a comparison of the instances in 
which a given phenomenon occurs, with the instances 
in which this phenomenon does not occur; and it is 
the sole circumstance in which these instances differ 
that is noted. The following is Mill's canon for this 
method: "If an instance in which the phenomenon 
under investigation occurs and an instance in which 
it docs not occur, have every circumstance in com- 
mon save one, that one occurring only in the former; 
the circumstance in which alone the two differ is the 
effect, or the cause, or an indispensable part of the 
cause, of the phenomenon." 

3. The Joint Method or Method of Double Agree- 
ment. — The distinctive feature of this method is the 
double employment of the method of agreement, this 
method being employed both in the instances in which 
the phenomenon occurs, and in the instances in which 
it does not occur. The method thus affords two dis- 
tinct proofs, each proceeding independently of the 
other, and each corroborating the other. The follow- 



1 68 ELEMENTARY LOGIC 

ing is the canon for this method: "If two or more 
instances in which the phenomenon occurs have only 
one circumstance in common, while two or more in- 
stances in which it does not occur have nothing in 
common save the absence of that circumstance, the 
circumstance in which alone the two sets of instances 
differ, is the effect, or the cause, or an indispensable 
part of the cause, of the given phenomenon." 

4. The Method of Residues. — This method is em- 
ployed in those cases in which some of the con- 
comitants of the phenomenon are already known 
to be causal antecedents and consequents; and the 
method consists in subtracting these from the totality 
of concomitant circumstances, so as to leave as the 
residuum, the causal antecedents yet to be ascer- 
tained. The canon of this method is, "Subduct 
from any phenomenon under investigation such part 
as is known by previous inductions to be the effect 
of certain antecedents, and the residue of the phenom- 
enon is the effect of the remaining antecedent." 

5. The Method of Concomitant Variation. — This 
method consists in ascertaining what variation in a 
given phenomenon occurs when a definite variation 
occurs in some other phenomenon. Its canon is, 
"Whatever phenomenon varies in any manner whenever 
another phenomenon varies in some particular manner, 
is either a cause or an effect of that phenomenon, or 
is connected with it through some effect of causation." 

The student will better understand the use of these 
methods if I add a few examples of their employment. 



OBSERVATION AND EXPERIMENT 169 

Elimination by Agreement. — I observe after taking a 
particular kind of food, I am invariably ill; a careful 
comparison of all instances in v^^hich this result follows 
shows that the taking of this kind of food is the only 
material circumstance in which they all agree; I infer 
from this fact that it is this kind of food that is the cause, 
or at least in part the cause, of my being made ill. 

Elimination by Difference. — A man, known to be in 
good health at a certain moment of time, falls dead; 
examination discovers that a bullet has penetrated his 
brain. A mass of gunpowder is in a magazine, a 
lighted match is put in contact with it, an explosion 
follows. The sole differencing circumstance in the in- 
stances of the man in health, and the man dead, was 
the bullet in his brain. Likewise in the* two in- 
stances, that of gunpowder in the magazine, and gun- 
powder destroyed by explosion, the sole differencing 
circumstance was the lighted match in contact with the 
powder. We say the bullet killed the man, and the 
match caused the explosion. 

Elimination by Double Agreement. — As an example 
of the employment of this method, I take the following 
from Fowler's " Inductive Logic " (p. 163). A ray of 
light proceeding from incandescent hydrogen is passed 
through a prism, and it is invariably found that, in 
the spectrum thus obtained, there are two bright lines 
occupying precisely the same position; moreover, rays 
of white hght proceeding from various incandescent 
substances are passed through incandescent hydrogen 
and the emergent light is then broken up by a prism. 



I/O ELEMENTARY LOGIC 

In the spectra thus obtained, it is found that there are 
invariably two dark (or under certain circumstances 
two bright) lines occupying exactly the same position 
in the spectrum. If we try the same experiments with 
any other elements than incandescent hydrogen, al- 
though we may obtain bright and dark lines, we never 
find these hnes occupying the same position in the 
spectrum as the two lines in question. 

As this case is not so simple as the ones given in 
illustration of the two first methods, I will analyze it. 
First, by the method of simple agreement it is shown that 
the ray passing through incandescent hydrogen and the 
invariable position of certain lines in the spectrum are 
causally connected things, since this passing through 
incandescent hydrogen is the sole agreeing circum- 
stance in the instances in which the phenomenon 
occurs. Secondly, it is shown by the same method 
applied to the negative instances that the absence of a 
ray passing through incandescent hydrogen is the sole 
antecedent on which the non -occurrence of this phe- 
nomenon is observed. 

Elimination by Residues. — The classic illustration of 
this method is the discovery of the planet Neptune. 
The facts are briefly these: Certain perturbations in 
the planet Uranus had been observed since 1804. It 
was known what amount of perturbation in the motions 
of this planet was due to the influence of known heavenly 
bodies. Deducting the effects of the known influence 
of these other bodies, there remained the perturbations 
for which a cause was to be discovered; and as the 



OBSERVATION AND EXPERIMENT 1 71 

Student probably knows, Mr. Adams in England and 
M. Le Verrier in France almost simultaneously calcu- 
lated the position of some planetary body which could 
occasion these disturbances in the motions of Uranus. 
Dr. Gill of the Royal Academy of Berlin turned his 
telescope to that region of the heavens, and discovered 
the planet Neptune. 

Elimination by Concomitant Variation. — A good 
example of this method are some observations upon the 
grip epidemic in New York, made by Weather Fore- 
caster Dunn. Mr. Dunn came to the conclusion, that 
humidity with change in temperature was the most im- 
portant element in causing the spread of the disease. 
The facts on which this inference was based are the 
following: (i) The fatahty was most marked when 
the humidity was at its maximum, and there was a 
sudden fall of the temperature. (2) The higher the 
humidity and the more sudden the fall of temperature, 
the greater was the number of deaths. (3) When, 
on the other hand, the temperature and the humidity 
dropped at the same time, there was a decrease in the 
death rate. 

A comparison between these methods may serve to 
bring out more closely their distinctive features. These 
methods, as has been shown, have a common function, 
that of eUminating the non- causally connected concom- 
itants of the given phenomenon. Each of these methods 
effects this ehmination in a different way. In most of 
the instances in which the causal antecedent is sought, 
more than one of these methods can be employed; 



172 ELEMENTARY LOGIC 

and, when this is the case, the evidence of causal con- 
nection is of course materially strengthened. Let us 
first compare the methods of agreement and differ- 
ence. 

A first point of difference between these methods is 
the principle on which each method proceeds. The 
method of agreement goes on the principle, that what- 
ever circumstance can be eliminated without affecting 
a given phenomenon, is not causally connected with 
this phenomenon; the principle of the method of 
difference is, whatever circumstance cannot be elimi- 
nated without affecting the given phenomenon, is a 
cause of this phenomenon. 

A second difference between these methods concerns 
the character of the instances with which each deals, 
and the way in which these instances are treated. The 
method of agreement requires us to observe or obtain 
by experiment instances which agree in but a single 
circumstance; the method of difference requires in- 
stances which agree in all the circumstances but one. 
Thus it is the agreeing circumstance that is important 
in the one method, while the differing circumstance is 
the important one in the other method; hence the 
names that aptly distinguish these methods. Again, 
notice that in the method of agreement comparison is 
made between all the instances in which the phenome- 
non occurs ; in the method of difference the comparison 
is made between an instance in which the phenome- 
non occurs with an instance in which this phenomenon 
does not occur. Only the presence of the phenomenon 



OBSERVATION AND EXPERIMENT 1 73 

is ascertained by the one method; both its presence 
and absence are ascertained by the other method. 

Elimination by Agreement and Difference. — This 
method differs from the method of agreement only in 
the circumstance that it takes account of negative as 
well as positive instances of the given phenomenon; 
that is, instances in which the phenomenon does not 
occur as well as those in which that phenomenon 
occurs. 

A comparison of this third method with the method 
of difference is not unimportant, because the student 
is hable to confound these methods, or at least to sup- 
pose that the method partakes of double agreement of 
the distinctive character of the method of difference. 
The name joint method is not so good a term as the 
other term, double agreement; because it impHes this 
mistaken connection between the two methods. It is 
true that the method of double agreement has this 
feature in common with the method of difference, viz. 
in it two sets of instances are observed, instances in which 
the phenomenon occurs, and instances in which it does 
not occur; but there is no further agreement between 
them ; on the contrary, there are these differences : — 

(i) In the method of double agreement it is the 
agreeing circumstance that is noted in both sets of in- 
stances; in the method of difference, it is the dis- 
agreeing circumstance that is noted. 
. (2) In the method of double agreement instances in 
which the phenomenon occurs are compared with each 
other, and instances in which the phenomenon does not 



174 ELEMENTARY LOGIC 

occur are compared with each other; in the method 
of difference the comparison is between instances in 
which the phenomenon does not occur, and instances in 
which it does occur. 

The method of residues when compared with the 
other methods presents these two pecuharities : — 

(i) It does not of itself estabhsh a causal connec- 
tion. It only eliminates known causally connected 
concomitants of a given phenomenon. The residual 
phenomenon with its concomitants is a problem to be 
solved either by the other methods, or in some instances 
by the method of hypothesis, as will be shown later. 
Thus, in the discovery of the planet Neptune all that 
was accomplished by this method was the separation of 
the given phenomenon and its concomitants into two 
parts, one containing antecedents and consequents 
known to be causally connected, the other containing a 
phenomenon and an unknown or unobserved cause; 
and it was by hypothesis and verification that this causal 
antecedent was discovered. The case would not have 
been materially different had this cause been among 
the observed concomitants of the given phenomenon; 
it would have been by the use of one of the other 
methods that this cause was discovered. 

(2) The second peculiarity of the method of residues 
is, that it is used to ascertain causal connection, not 
only between observed phenomena, but between an 
observed phenomenon and something that is not ob- 
served; while the other methods are limited to causal 
connections between observed phenomena. 



observation and experiment 1 75 

Section 30 

the logical value of the methods of observa- 
tion and experiment 

I shall discuss in this section the evidence of causal 
connection which these methods afford. Let us first 
assume that the situations which these methods pre- 
suppose are actual and reahzable; we will assume 
that these methods can be ideally carried out, that 
such instances as the method of agreement, for instance, 
contemplates are met with in experience. 

Even under such ideal conditions as we have supposed, 
these methods come short of satisfying the canons of 
formal logic. It is quite certain that if ^ is a cause 
of a given phenomenon, A will always be present 
when that phenomenon occurs; but, to infer from 
the uniform presence of ^ as a circumstance that is 
always present that it is a cause, is to commit the 
fallacy of affirming a consequent. Judged, therefore, 
by the canons of formal logic, these methods do not 
make us logically certain of causal connection. But 
if the evidence possible by these methods falls short of 
certainty, it can and does approximate that ideal of 
evidence. Between the probabihty of causal connec- 
tion which these methods, even under actual condi- 
tions, attain and certainty there is no difference of any 
practical value. We attain to a conviction that is so 
practically sufficient, and so rationally satisfying, that 
no really sound mind feels an inchnation to doubt, or 
could justify itself in so doing. 



176 ELEMENTARY LOGIC 

There is, however, great inequahty in these methods 
in respect to their evidential value. The weakest of 
them is the method of agreement; and the strongest, 
the method of difference. Perhaps next in evidential 
value should be placed the method of concomitant 
variations. These methods, as we have observed, 
can very considerably corroborate each other; since 
in most cases more than one of them can be employed. 
The method of agreement is relatively weak for the 
reason, as Mill observes, that it at best only establishes 
the presence of a particular circumstance when a given 
phenomenon occurs; it cannot make us certain that 
other circumstances are not also present but unobserved ; 
nor can it make us certain that if this particular cir- 
cumstance were not present, the phenomenon would 
not occur. Let us suppose that A is always present 
when B occurs, and in fulfillment of the requirements of 
this method, A is the only observed circumstance that 
is always present when B occurs; this situation per- 
mits no less than four inferences: 

(i) ^ and B are related as cause and effect. 

(2) Both A and B are effects of some unobserved 
cause. 

(3) A, though always present, is not itself the cause 
of B, but the cause of some circumstance, either ob- 
served or unobserved, which is causally connected with B. 

(4) A, though the cause of B, is not the sole cause; 
some other circumstance present would be the cause 
of B in the absence of A , this other circumstance being 
latent owing to the influence of A. 



OBSERVATION AND EXPERIMENT 1 77 

Thus our first method comes short of establishing 
causal connection. The method of difference is far 
more cogent in the inference it warrants. Mill accords 
to this method the highest degree of evidence, amount- 
ing to practical demonstration even under actual con- 
ditions. "It is by the method of difference alone that 
we can ever, in the way of direct experience, arrive with 
certainty at causes" ("Logic," p. 282). The reason for 
the greater cogency of this method. Mill finds in the 
fact that the nature of the combinations which it 
requires is much more strictly defined than in the 
method of agreement. 

This method, requiring that the two instances be 
alike in all circumstances save one, and also involving 
both presence and absence of the given phenomenon, 
is a much more effective instrument for the elimination 
of non-causal concomitants than is any one of the 
other methods. But what most of all makes this 
method the strongest of the methods is the fact 
that it permits a completer employment of experiment ; 
it is preeminently an experimental method, and 
therein lies its effectiveness. It is thus possible to 
introduce into a given state of circumstances a change 
that is of a perfectly definite nature, and to observe 
what results. To quote again from Mill (" Logic," p. 
281 : ) "We choose a previous state of things with which 
we are well acquainted; so thai no unforeseen altera- 
tion in that state is hkely to pass unobserved ; and into 
this we introduce, as rapidly as possible, the phenome- 
non which we wish to study ; so that in general we are 



178 ELEMENTARY LOGIC 

entitled to feel complete assurance, that the preexisting 
state and the state which we have produced differ in 
nothing except the presence or the absence of this phe- 
nomenon." " If a bird is taken from a cage, and plunged 
into carbonic acid gas, the experimenter may be fully 
assured that no circumstance capable of causing suffo- 
cation has supervened in the interim except the change 
from immersion in the atmosphere to immersion in 
carbonic acid gas." 

To a considerable degree what is true of this last 
method is true of the method of concomitant varia- 
tion; this likewise admits of the use of instruments 
of exact measurement; and where such instruments 
can be employed, it is possible to establish relations 
that are so definite in character as to make the 
inference of causal connection scarcely less compel- 
ling than is the evidence afforded by the method of 
difference. Since it is upon the definite character of 
the variations that this method rehes, its evidential 
value is proportionate to the degree of definiteness 
that these variations present. Now, when it is pos- 
sible to establish mathematical relations, such as ratios, 
relations of weight, volume, intensity, motions, etc., 
the evidence of causal connection thus afforded, it will 
be readily perceived, is very strong. 

The method of double agreement, owing to the 
negative instances it considers, has greater evidential 
force than does the method of single agreement; but, 
inasmuch as it does not involve a comparison between 
positive and negative instances, and does not make 



OBSERVATION AND EXPERIMENT 1 79 

possible the use of experiment to such an extent as the 
method of difiference, this method is distinctly weaker 
than the former. 

In this estimation of the logical value of these methods 
we have assumed that they are employed under con- 
ditions that completely satisfy their requirements. 
Experience teaches, however, that such conditions are 
in no cases afforded us. It is this discrepancy between 
the hypothetical conditions of these methods and the 
actual conditions to which they are apphed that con- 
stitutes the inherent weakness of them all ; though this 
weakness affects some of them to a greater degree than 
it does others. The possibihty of there being unob- 
served concomitants of the given phenomenon despite 
our most careful analysis; the possibihty that more 
than one circumstance stands in causal connection with 
the given phenomenon; the possibihty that the inva- 
riable coexistence of the two phenomena in our rather 
limited experience may be a non-causal coincidence 
only, — these possibihties, it must be confessed, weaken 
the evidence of causal connection which in very many 
cases it is possible for us to obtain. 

The complexity of nature is too great, and our powers 
of analysis and accurate observation are too Umited, 
to enable us to attain more than a reasonable prob- 
abihty that we have discovered causal connection 
between phenomena, that we have successfully traced 
out the numerous and intricate threads of causal 
hnkage that compose the vast web of nature. Still 
more is this true of the phenomena of human actions, 



l80 ELEMENTARY LOGIC 

— social phenomena. To discover, in this realm, laws 
of causal connection is a goal of endeavor yet far 
in advance of any present achievement. Observation 
here being our main rehance, and the method of 
agreement the only one that in many cases is admis- 
sible, we can understand how precarious are the con- 
clusions in most reasonings in the so-called moral 
sciences. 



CHAPTER XII 

EXPLANATION BY HYPOTHESIS 

All real explanation in science proceeds by hy- 
pothesis and verification. The ascertainment of causal 
connection by the methods just described is only a 
preliminary step toward explanation. The causal 
connections themselves only give the phenomena to 
be explained that orderly character, that form in which 
they can become the data for scientific explanation. 
These causal connections themselves become new prob- 
lems for explanation. 

Hypothesis is the great instrument of science. Every 
important advance in man's knowledge of the universe 
has involved this step from the known to the unknown. 
Nothing is so characteristic of the great minds in sci- 
ence, as the ability and the courage to make hypotheses 
and rigorously to test them. It is for this reason that 
scarcely any quality is more requisite to the investigator, 
the discoverer, than what Tyndall happily calls the 
scientific imagination. Original men have possessed 
this faculty in a high degree. To this r61e of the imagi- 
nation in science, we owe all the great discoveries, 
the brilhant achievements, and the most successful 
working hypotheses that have distinguished the great 



1 82 ELEMENTARY LOGIC 

century now passed. Hypothesis has been already 
defined, and its function in a general way explained. 
I shall now proceed to an exposition of the use of 
hypotheses in science. 

Section 31 

the essential features of a scientific 
hypothesis 

The Requisites of a Legitimate Hypothesis. — Be- 
cause hypothesis is a step beyond the known, beyond 
the solid ground of experienced facts ; and because this 
step is a venture in which imagination carries us in one 
sense to worlds unknown, it would be very erroneous to 
infer that any sort of step, any kind of venture or flight 
of imagination, is permitted in science, provided in some 
way we can get back again to our actual world. Fic- 
tions are permitted in science, but only such fictions as 
help to the understanding of facts, only such fictions 
as enable us to link facts into an orderly, a coherent 
and rational universe. Hence, the requisites of a per- 
missible hypothesis are : — 

(i) That it conforms to the analogies of experience. 
By this I mean that whatever agent or mode of action 
of an already known agent is supposed, it must not be 
so unlike that which we already know in experience that 
it cannot be clearly conceived in terms of our experience. 
Knowledge involves a step to the unknown; but this 
step cannot be an absolute break in the continuity of 
thought and possible experience, or we are left as 



EXPLANATION BY HYPOTHESIS 1 83 

ignorant as we were before taking this step. There 
must be some term of relation between that which the 
hypothesis supposes and the datum from which it 
starts; and no term of relation is possible between 
what is absolutely unknown and our known world. 
Science, therefore, permits the construction of no hy- 
potheses which involve the conception of something 
totally unlike that which we already know. 

(2) The second requisite of a legitimate hypothesis is 
the possibihty of deducing from it phenomena of ex- 
perience ; and this deduction must be based upon rela- 
tions that are rational. Any other hypothesis violates 
a fundamental condition, viz. that it shall explain, that 
it shall lead back to the known. A merely supposed 
something from which we can get nothing in the way 
of rendering given phenomena more intelligible than 
they are already, merely mocks us with the semblance 
of explanation. The necessary assumption on which 
science proceeds is the essential, the rational, continuity 
of that which is not yet known, with the facts of experience. 

An hypothesis is a thought construction by the aid 
of which we make this continuity definite and sensibly 
realizable; its function is thus to extend our world in 
terms of possible experience. Hence, no hypothesis is 
permissible that sets up objects which are unrelated to 
the objects of actual experience ; and from which, con- 
sequently, we cannot deduce the objects and processes 
of which the world of experience consists. 

The scientific imagination, it must be remembered, 
is imagination working under the control of reason and 



1 84 ELEMENTARY LOGIC 

for the ends of knowledge. Science permits this voyage 
upon unknown seas and to unknown lands; but only 
if it be no idle or aimless venture, but a serious quest of 
truth, a voyage of possible discovery for the purpose 
of enlarging our knowledge and our practical control 
of nature. 

Section 32 

the method of explanation by hypothesis 

Four steps have been distinguished in this method: 
(i) Constructing the datum; 

(2) Constructing the hypothesis; 

(3) Deducing the consequences from the supposition ; 

(4) Comparing these consequences with facts of expe- 
rience. 

Induction, hypothesis, deduction, and verification are 
the technical names for these four steps. The first of 
these processes, however, does not properly belong to 
explanation, because explanation presupposes the datum 
already defined or construed; and this construing or 
definition of the datum belongs to the stage in science 
we have already described, — that of observation and 
experiment. The problem must first be accurately 
stated before a solution is undertaken ; and hypothesis 
is just a method of solving a given problem. 

Nor is it advisable so to distinguish deduction and 
verification as to make these processes separate steps; 
these are only distinguishable elements in one pro- 
cess, which is verification, or proof of the hypothesis. 



EXPLANATION BY HYPOTHESIS 1 85 

Accordingly, there are two and only two stages or steps 
in explanation by hypothesis; these are: (i) the con- 
struction of the hypothesis, and (2) its verification or 
proof. We shall now discuss these processes. 

I. The situations in which we construct hypotheses 
are various; they range from the simplest facts of ob- 
servation which daily hfe presents, to those situations 
in which, by the regulated methods of observation and 
experiment, order and causal connection have been 
established among the more complex phenomena of 
experience. Whatever be the situation or the char- 
acter of the phenomena that confront us, the essential 
operation of constructing an hypothesis is the same; it 
is a conjecture, a conception of some agent or mode of 
action, by means of which the new facts can be as- 
similated to what we already know, can be fitted into 
a coherent and mentally satisfying experience. Intel- 
lectual perplexity in the presence of given phenomena, 
uneasiness and dissatisfaction until this perplexity is re- 
moved by rational explanation, are the impelHng motives 
to all hypothesis building. 

To the formation of a good and serviceable hypothe- 
sis, two things are requisite, — accurate knowledge 
of the given facts, and analogical suggestion; by the 
latter I mean the detection of significant agreements 
between given facts and other facts already explained. 
I shall have occasion in another place to discuss the 
nature and value of analogical inference; here, I will 
only remark that its scientific value lies chiefly in its 
suggestion of hypothesis. Indeed, analogical suggestion 



1 86 ELEMENTARY LOGIC 

is the basis on which every hypothesis rests ; we might 
say the invention of an hypothesis is an analogical 
inference. 

2. The second step in explanation by hypothesis is 
verification, which consists of a statement of what 
phenomena ought to be observed if the hypothesis is 
true, and a comparison between these phenomena and 
those of actual experience. In this deduction of con- 
sequences and their comparison with actual facts con- 
sists the test of the hypothesis. These two moments 
in the process of verification can be expressed as the 
premises of a hypothetical syllogism ; the major premise 
is the statement of the hypothesis and that which is 
deduced therefrom, and the minor premise states the 
result of the comparison with experience, either by 
asserting that there is, or asserting that there is not, such 
agreement between the hypothesis and the facts of 
experience. When the phenomena deduced from the 
supposition agree with experience, the hypothesis is 
said to be verified. If this agreement is complete, so 
that there are no facts left unexplained, the hypothesis 
is said to be completely verified, sometimes said to be 
true or proved. Incompleteness of verification, it is 
obvious, is a matter of degree; the partial verification 
may leave considerable areas of fact unexplained, and 
yet be an admissible and serviceable hypothesis. 

Two distinctions are of sufficient importance to de- 
serve attention at this point. They are : — 

(i) The distinction between complete verification and 
complete proof, and 



EXPLANATION BY HYPOTHESIS 1 87 

(2) Between incomplete verification and disproof of 
an hypothesis. 

It is necessary to the complete proof of an hypothesis 
that it should be completely verified; but a com- 
pletely verified hypothesis is not thereby completely 
proved. An hypothesis is completely proved only 
when that which was supposed, is otherwise discovered 
to be a known fact of experience, or when that hypothe- 
sis is demonstrated to be the only one which can ex- 
plain the given phenomena. To be completely proved, 
then, an hypothesis must either cease to be mere hy- 
pothesis and pass into fact, or, remaining an hypothesis, 
be the only possible one in the given situation. 

An instance of a completely proved hypothesis was 
the discovery of the planet Neptune. Up to the hour 
when the telescope revealed that body, Neptune was 
merely a supposed being, a completely verified hypothe- 
sis, to be sure, but still an hypothesis only. The 
revelation of the telescope completely proved that 
hypothesis, and it did so by converting hypothesis into 
fact; Neptune became a known fact of experience. 

To illustrate the second condition in which an hy- 
pothesis is completely proved, let us suppose that the 
hypothesis was that of a heavenly body so situated in 
space that no telescope had as yet found it ; if, under 
that condition, this supposed body was the only one 
which could explain the disturbances in the motions of 
Uranus, astronomers would have been as certain of its 
existence as they were of the existence of Neptune after 
the telescope had brought it into the sensible world. 



1 88 ELEMENTARY LOGIC 

But a sole hypothesis, such as complete proof presup- 
poses, is an ideal, not an attained actuality. It may be 
shown that the given hypothesis is the only one yet 
proposed that explains the phenomenon under investi- 
gation ; but to demonstrate that no other hypothesis is 
conceivable or v^^ill ever be framed which can explain 
that phenomenon is something our human minds, sub- 
ject to change and growth, cannot do. 

The most that can be claimed for any hypothesis is 
that it is so far as known the only one that is admis- 
sible ; and this approximation toward a sole hypothesis 
constitutes very strong evidence, stronger, of course, 
than the evidence afforded by complete verification; 
but it is evidence that falls short of complete proof. 
There is, consequently, but one way in which an hy- 
pothesis is completely proved ; and this absolute proof 
at the same time transforms the supposed, into known 
reality. 

The other distinction, that between partial verifica- 
tion and disproof, must not be overlooked if we would 
understand the actual procedure in science. A per- 
fectly admissible and even very serviceable hypothesis 
may be one which leaves a part of given phenomena 
unexplained; but no hypothesis is admissible after it 
is shown to be in contradiction with any one fact of 
experience. One contradictory fact is a disproof of an 
hypothesis, while many unexplained facts are not in- 
compatible with a tenable and useful hypothesis. 

These distinctions explained, we pass now to a dis- 
cussion of the evidence for hypothesis. We begin with 



EXPLANATION BY HYPOTHESIS 189 

the evidence afforded by complete verification. Judged 
by the canons of the hypothetical syllogism, such com- 
plete verification does not establish the truth of the 
hypothesis; indeed, to accept such an hypothesis as 
true commits the fallacy of affirming the consequent. 
But this discrepancy between the evidence demanded 
by the hypothetical syllogism and the evidence that is 
available for hypotheses in science only serves to accen- 
tuate that difference we have insisted upon between 
formal logic and reasonings upon matters of fact. The 
syllogism, as has been shown, recognizes no conclu- 
sions that are not certain, and admits no proof that does 
not estabhsh such conclusions. In matters of fact, 
probabihty is all that the best evidence obtainable can 
give us. Nor, do the principles which are regulative 
for syllogistic reasoning afford any standard by which 
the evidence for scientific hypotheses can be tested and 
measured. 

The grounds on which the scientific mind accepts 
any particular hypothesis cannot be formulated in the 
terms of formal logic. In the world of concrete facts 
other principles are regulative for our thought; and 
there are other criteria of rational belief than those 
which belong to the world of mere conceptions. Expe- 
rience is not only the datum or starting point for all 
scientific reasoning, but also the test by which the evi- 
dential force of all such reasoning is to be measured. 
To work completely in experience, to make experience 
inteUigible, coherent, and practically, as well as theo- 
retically, satisfactory, is the criterion of a true concep- 



IQO ELEMENTARY LOGIC 

tion or hypothesis. Does an hypothesis enable us to 
comprehend the given phenomena, does it enable the 
mind to forecast them, does it enable us to know what 
to expect and what practically to prepare for, so that 
our mental prevision and our actions as well are de- 
fined and made sure ? These are the tests by which an 
hypothesis is judged; and these afford the standard by 
which the evidence for it is measured. 

Next, therefore, in credibility to the sole hypothesis, 
we should ordinarily rank the completely verified 
hypothesis; while hypotheses that are only partially 
verified would be weakest in point of evidence. But 
there is one circumstance which affects the relative 
evidential value of the completely verified and the in- 
completely verified hypotheses, — it is the character of 
the phenomena they explain. There are conditions 
under which an hypothesis which is only partially 
verified is accepted with more confidence than a com- 
pletely verified hypothesis is under other conditions. 
An hypothesis may be completely verified within a 
relatively narrow range of phenomena, and with phe- 
nomena of simple character, which is not entitled to 
so much confidence as another hypothesis which is 
incompletely verified, but is applicable to a ver}^ wide 
range of phenomena, or to a very complex and pecuhar 
phenomenon. Accordingly, the evidential value of an 
hypothesis is, in part, determined by the extent or unique 
character of the phenomena to which it is applied. 

Summing up this discussion upon the evidential value of 
hypotheses, we can formulate the chief points as follows : — 



EXPLANATION BY HYPOTHESIS 191 

(i) Next to a completely proved hypothesis the one 
entitled to most confidence is the hypothesis 
which, within the limits of what is known, is the 
only one that can explain the given phenomenon. 

(2) Other things being equal, the completely verified 

hypothesis ranks next in credibility. 

(3) Other things being equal, that hypothesis is most 

credible which is applicable to the widest range 
of phenomena, or to phenomena most complex 
in character. 

(4) An hypothesis is admissible, provided it explains 

any part of the given phenomenon and is con- 
tradicted by no phenomenon. 

(5) An hypothesis is no longer tenable when a single 

phenomenon is found which contradicts this 
hypothesis. 

Section 33 

the value of rejected hypotheses 

It is not alone the true hypotheses which are useful 
to science; science is also indebted to her rejected 
hypotheses; each one of these has done something to 
prepare the way for the more successful ones that have 
taken their places. The observation, often made, that 
the pathway of science is strewn with the wrecks of 
exploded theories, discarded hypotheses, involves an 
erroneous conception of the nature and growth of our 
human knowledge. Truth is not reached by a single 
bound. Seldom has one leap of the scientific imagina- 



192 ELEMENTARY LOGIC 

tion brought the mind to the right, or at least to the 
final, explanation. 

Hypotheses in science become established as do 
species in the organic kingdom, — through the struggle 
for existence and the survival of the fittest. But the 
unsuccessful and the defeated competitors have contrib- 
uted to that progress which has left them behind to 
perish. Hardly has there been an hypothesis so erro- 
neous as not to contain some element of truth, and which 
has not in consequence of that truth helped the estab- 
lishment of a truer one. It is by the testing and 
rejection of the bad hypotheses that the conditions 
become better defined, the problem more correctly 
stated, and the requisites of the true hypotheses more 
accurately apprehended. 

The pathway of all our knowledge leads through 
errors and partial failures. Science has won her 
progress through half truths and rejected untruths; 
she has advanced not alone by successful steps, but by 
mistaken and corrected steps. The men who have been 
most successful in science are those who have most 
generously recognized the measure of truth in con- 
ceptions they have rejected, and who have taken most 
pains to understand doctrines which the progress of 
science has made no longer tenable. The truth is the 
whole ; and the sure mark of the conception that can 
claim final truth is its capacity to fulfill, and not to 
destroy, the conceptions it displaces. 



CHAPTER XIII 

THE THIRD SPECIAL PROBLEM IN THE 
LOGIC OF SCIENCE 

Section 34 
the calculation of chances 

As we have seen, the phenomena which give rise to 
this third problem are of two sorts : — 

(i) Individual events whose occurrence can be pre- 
dicted with varying degrees of probabihty ; and 

(2) Phenomena which, considered as aggregates or 
masses, present such uniformities as to admit of pre- 
diction with a high degree of probabihty. 

We shall now consider the phenomena of the first 
description. 

The terms chance and probability do not signify a 
quahty of any phenomena or events in themselves con- 
sidered. There are no chance or probable events of 
that sort in the real world; such an event would be 
one which had no reason for its occurrence, and which 
was consequently without connection with any other 
phenomena. Such an event is absolutely unthinkable 
and rationally impossible. These terms describe cer- 
o 193 



194 ELEMENTARY LOGIC 

tain states of mind or mental attitudes toward certain 
events in our real world ; in other words, chance and 
probabihty describe states of our minds and not quahties 
of any events in nature. They are confessions of igno- 
rance, the limitation of our knowledge. 

Did we completely know our world, such mental 
attitudes would be impossible. But, on the other hand, 
were we utterly ignorant, there would be neither chance 
nor probability ; it is because we are both knowing and 
ignorant that we can speak of some things as chance 
events, and regard their occurrence as probable. A 
chance event is an event of whose cause we are 
ignorant. A probable event is an event for the occur- 
rence of which some reasons exist in what we know of 
it, or of the class to which it belongs. 

Probability properly means any conviction that is 
less than certainty, and a conviction the evidence or 
reason for which admits of estimation or measurement. 
When, for example, I say it is probable that the war in 
the East will soon be brought to an end, I express a 
conviction of a certain measurable degree. If, in addi- 
tion, I say the probabilities are four times as great that 
Japan will come off victorious as are the probabili- 
ties that Russia will triumph, I give to my convic- 
tion a definite measurement; I express its relative 
strength in quantitative terms. To be more accurate, 
I mean by this statement that there exist for my mind 
four times as many reasons, or four times as strong a 
reason, for expecting the success of Japan as there are 
for expecting the success of Russia. 



THE THIRD SPECIAL PROBLEM 195 

The importance of making clear the exact meaning 
of chance and probabiHty justifies our hngering longer 
upon this topic. Let us, accordingly, note the situa- 
tions in which we can properly characterize a contem- 
plated event as one of chance occurrence. That situa- 
tion is clearly one in which we must know something — 
we must know that some event is to occur; but, at the 
same time, we do not know what particular event, or 
what is the particular description of the event that is 
to happen. In other words, we know that one of sev- 
eral possible events is to be actual ; but we do not know 
which one of these possible events is to become the actual 
one. 

Let me describe this situation in somewhat differ- 
ent terms. It is a situation in which we know that more 
than one event is possible. It is a situation in which we 
know that one event at least will be actual. It is a 
situation in which we have no better reason for expect- 
ing one of the alternative possible events, than for ex- 
pecting any other one of them, our ignorance of causes 
being equally distributed among the possible events. 
Now, that state of mind in which I entertain the occur- 
rence of any one of those possible events is what 
we should mean by chance. If I regard the strength 
or degree of conviction with which I expect the occur- 
rence of any particular event, that mental state is what 
should be meant by probability. 

Thus, chance and probabihty can characterize the 
same event, — it is a chance event, because its particu- 
lar cause is not known ; it is a probable event, because 



196 ELEMENTARY LOGIC 

its occurrence is expected with a given degree of con- 
viction. 

And this leads to our topic, The Calculation of 
Chances. What is it to calculate chances and what 
are the methods of such calculation? Our best way 
of answering these questions is to examine some con- 
crete cases in which this is done. Let us take as the 
first case, throwing a die. Antecedent to the throw, 
it is known that one of the six faces must come upper- 
most; the structure of the die makes this fact certain. 
We know also that some cause will determine a particu- 
lar side to be uppermost ; but, as we do not know what 
that cause is, the reasons we have for expecting one side 
to come uppermost are no greater than are those for 
expecting any other one of the six sides to be uppermost. 
We therefore say the chances are equal. Because 
these chances are equal, and because there are six pos- 
sible events, the chances that it will be a particular side, 
say a six-spot, that will be uppermost, are but one fifth 
as many as the chances that favor any one of the re- 
maining five sides ; or the chance of a six-spot coming 
uppermost is one in six of the total chances. Expressed 
in other terms, our calculation in this instance is, the 
chances against my throwing a six-spot are five to one. 
Now, the analysis of this case shows that a strictly 
mathematical measurement applies to the reasons there 
are for expecting a particular event, — the six- spot side 
coming uppermost. 

This mathematical calculation assumes two 
things : — 



THE THIRD SPECIAL PROBLEM 



197 



(i) That our minds are not affected by any experience 
of the results from prior throws; 

(2) That we are equally ignorant of the special 
causes which will determine any one of these sides to be 
uppermost, these reasons being simply counted, and not 
estimated or weighed. 

As a second case, let me suppose a box containing 
black balls and white balls mixed in a proportion which 
I do not know. Now, in such a situation, antecedent 
to my drawing a ball we should say the chances are 
equal, and the probability that I shall draw a white 
ball is one to two, or |. Now, let us suppose the box con- 
tains white and back balls, in the proportion of twenty 
white balls to five black balls ; under these conditions, 
my expectation of drawing, say a black ball, is express- 
ible by the fraction ^-^ or \. 

In the two cases now examined, we have a strictly 
mathematical measurement, or calculation of chances. 
This is so because a definite number of events is con- 
sidered, and each one of these events, so far as we 
know, is equally possible; and because no other circum- 
stance influences our judgment. 

The next case presents a somewhat different situa- 
tion. It is again a box containing balls; but now, 
neither the number nor the color of these balls is 
known. In this situation it is not possible to calculate 
chances prior to a result ascertained by drawing. I 
can entertain no expectation respecting the sort of 
ball that is to come out at the first draw. Now, 
suppose I draw six white balls in succession, what 



iqS elementary logic 

are the chances respecting my next draw? Have I 
more reason to expect a white ball next time than for 
expecting some other color? May I presume that the 
box contains only white balls, and if so, can the degree 
of this presumption be mathematically estimated ? 

Before answering these questions, let us note care- 
fully the difference between this case and the two preced- 
ing ones ; this difference is the circumstance, that in the 
first two cases the calculation of chances was made prior 
to and independently of experience ; while in this case, 
it is the result of the successive drawings that forms the 
basis of a calculation of chances, if such a calculation is 
admissible. In those cases the influence of experience 
was precluded; in this case experience is the sole 
determiner of my state of mind. 

Does this experience supply a basis for the same 
sort of calculation as that we made in the preceding 
instances? There is no question that this uniform 
experience of drawing only white balls justifies an ex- 
pectation that the next draw will give a white ball rather 
than a ball of any other color; the only question is, 
Can we give a mathematical expression to this expect- 
ancy or probabihty as we did in the other cases ? This 
question is to be answered in the affirmative and for 
this reason ; — after the six successive drawings of white 
balls and before the seventh drawing, we have essen- 
tially the same situation that the other cases presented. 
Having drawn six white balls and no balls of another 
color, I may assume that the white balls are six times 
as many as the balls of other colors ; and, consequently 



THE THIRD SPECIAL PROBLEM 199 

in the seventh drawing, the chances of a white ball com- 
ing out are expressed by the fraction f ; and of a ball 
that is not white, y. 

Let us suppose that of the six drawings, four have 
given white balls and two red balls ; then, in the seventh 
draw, the probabihty of a white ball coming out would 
be expressed by the fraction ^, and the probabihty of a 
red ball by the fraction f ; while the probabihty of 
a ball of some other color would be indicated by the 
fraction ^. 

In the calculation of chances we distinguish two sorts 
of cases: (i) those in which the number of alternative 
possible events is determined by the known conditions 
under which these events must occur, — throwing dice, 
drawing cards from a pack, etc., are instances of this 
class of cases; and (2) those cases in which the occur- 
rence of a phenomenon a certain number of times, in 
succession, either without or with interruption, is the 
basis for calculating the chances of the next occurrence ; 
our third case is an instance of this class. 

The calculation of chances we have considered relates 
to a single occurrence of the same event ; the calculation 
cannot be the same for more than one occurrence of 
the same event in succession. For instance, the prob- 
abihty of throwing a six-spot twice in succession is not 
one half as great as the probabihty of a single occur- 
rence of this sort, but is expressed by the fraction -^q, 
and the probability of getting a six spot three times in 
succession is gie* Hence, theoretically regarded, there 
is a rapid diminution of the chances favoring the repeti- 



200 ELEMENTARY LOGIC 

tion of the same event in successive instances. This 
decrease is not in terms of an arithmetical series, but 
in terms of a geometrical series. From this exposition, 
we derive the following rules for the calculation of 
chances : — 

(i) For a single occurrence of the specified event in the 
first class of cases, the probabiHty is expressed by 
a fraction having for its numerator, unity, and 
for its denominator, the number of possible 
events considered. 

(2) For a single occurrence of a specified event of the 

second class, the probabiHty is expressed by a 
fraction having for its numerator the total number 
of times the specified event has occurred, and 
for the denominator this number increased by 
one. 

(3) For the occurrence of the same event more than 

once, the probabiHty is expressed by a fraction 
whose numerator is unity, and whose denomi- 
nator is the number of possible events raised to 
the power denoted by the number of times the 
given event is to occur in succession. 
In our exposition of the doctrine of chance thus far, 
we have assumed that the theoretical calculations accord 
with the actual results obtained. There is, however, 
a discrepancy between theory and fact in the so-called 
chance events; and the problem presented by this dis- 
crepancy is to determine the amount of discrepancy 
that is compatible with the chance character of these 
events. This problem is better stated, perhaps, by this 



THE THIRD SPECIAL PROBLEM 201 

question : After how many repetitions of the same event 
may we infer that there is some special cause operating 
to produce this event? or, When are we justified in 
beheving that the coincidences are more than should 
occur, if they were only chance coincidences ? 

Numerous experiments have shown that the theoret- 
ical and the actual results approach agreement as 
the number of trials is greatly increased. For instance, 
one hundred throws of a penny gave seventy heads and 
thirty tails, but in upward of five thousand throws the 
number of heads closely approximated the theoretical 
number. Similar results have been obtained from a 
very protracted series of drawings of cards from a pack. 
These experiments indicate that the theory of prob- 
abihty holds true, if a sufficiently long run of instances 
is obtained. 

Now, it is just this fact of a discrepancy between 
theoretical results and actual ones in the hmited series, 
but which tends to disappear as the series is prolonged, 
that gives significance to our question, — When does 
this excess of actual over theoretical coincidences 
justify the behef that some particular cause is operative 
in producing this result ? For instance, after how many 
heads in succession should I be justified in believing 
that the penny is one-sided ? or after how many six-spots 
uppermost should I believe that the die is loaded ? 

This question hardly admits of a definite answer, so 
much depends upon the character of the phenomena 
and the conditions of their occurrence. I might be 
justified in believing that something more than chance 



202 ELEMENTARY LOGIC 

coincidence exists in one case, while in another case the 
excess of actual over theoretical coincidences would not 
justify such a behef. With such simple phenomena as 
tossing a penny, or throwing a die, or drawing a ball 
from a box, one seems justified in inferring a special 
cause when the coincidences are much in excess, and 
when they persist rather than diminish as the experi- 
ments are continued. But with more complex phe- 
nomena, the conditions of which are obscure, such a 
belief would not be justified. 

As a matter of fact, the theory of probabihty has little 
influence upon our behefs regarding future events. We 
do not regulate or measure our expectations of par- 
ticular events, whose causes we do not know, by the 
rules for calculation of chances. Experience and our 
knowledge of similar cases determine mainly and 
properly our behefs. If I throw a six-spot twice in 
succession, it does not seem improbable that the next 
throw will give a six-spot also ; but it would seem very 
improbable that I should get this result five times in 
succession. The reason for this difference in my ex- 
pectation is, that experience has shown that two sixes 
in succession is not uncommon, but a succession of five 
sixes is very uncommon. Those who are adepts in 
games of chance regulate their ventures by what they 
have found to be the habits of these phenomena ; what 
is called a run of luck, or the tide, are these habits which 
seem to belong to all phenomena. 

The calculation of chances, however, is not a useless 
method, simply because it does not regulate our prac- 



THE THIRD SPECIAL PROBLEM 2O3 

tical beliefs; this treatment of phenomena possesses 
a scientific value; because, by it the way is prepared 
for investigations that lead to the discovery of causal 
uniformities and for the use of hypotheses. The cal- 
culation of chances is sometimes a first step in bring- 
ing a group of phenomena, or a special phenomenon, 
within the domain of explained facts. The calcu- 
lation of chances is thus a method of more accurate 
description of certain phenomena ; and this more ac- 
curate description is an indispensable preliminary to 
scientific explanation. 

Section 35 
the method of statistics 

The second class of unexplained phenomena com- 
prises those events which present certain uniformities 
when considered in considerable numbers or masses, 
and when observed through considerable periods of 
time. It must be carefully noted that these uniformities 
hold true, not of the individuals that constitute these 
aggregates, but of the aggregates as such. Thus, when 
it is said that in a given population, say ten thou- 
sand, the mean death rate is j^, 1 : 100, it is not 
meant that this uniformity holds true of persons taken 
singly or individually, but that out of this given aggre- 
gate the mean or average number of deaths is one 
hundred. 

By the mean or average number in this method 
is meant that number which remains relatively con- 



204 ELEMENTARY LOGIC 

stant during a given period of time ; and by the mean 
or average individual or person is meant that fictitious 
individual who presents those characteristics which are 
selected as descriptive of this group. Thus, if I say 
the average German is five feet and eight inches in 
height, I do not mean that any one actual individual 
German is just five feet eight inches in height, but I 
mean that, could the height of every German be made 
equal, that height would be five feet eight inches. This 
mean or average German is therefore a fictitious indi- 
vidual. With these explanations we pass to our topic, 
The Method of Statistics. 

Statistics are any facts which are ascertained for a 
specific purpose; thus, to gather statistics relative to 
the effect of a certain occupation upon the health of 
those who are engaged in it, means that one ascertains 
such facts as the following: the number of persons 
engaged in this occupation, their ages, their sex, the 
localities in which they Hve, etc. Statistics are thus 
selected facts, facts of a definite character, and always 
numerically defined, and always for a purpose pre- 
viously determined. This method includes two opera- 
tions : — 

(i) Gathering statistics; 

(2) The ascertainment of uniformities and mean aver- 
ages presented by the phenomena thus specifically 
grouped. For example, statistics relating to accidents 
by railway travel are gathered. From these as 
data the average number of accidents for a year is 
ascertained, and the ratio of this number to the num- 



THE THIRD SPECIAL PROBLEM 205 

ber of persons who travel during this time. And, 
finally, it is shown that this mean ratio remains ap- 
proximately constant during a period of years. Thus 
a uniformity for a class of phenomena is estabhshed. 

The value of this method of dealing with phenomena 
is mainly practical, though its scientific value is not 
unimportant. The practical service of this method 
is illustrated by some of the most important business 
organizations or organizations for social improvement. 
The great organizations of life and property insurance 
have their foundation in those uniformities which this 
method has ascertained, and derive their stabihty 
from the constancy of the mean ratios, shown in the 
actuary tables of these companies. Observation 
covering a long period of years has shown that the 
causes, whatever they may be, which produce death, 
injuries, and the destruction of property, so operate 
as to produce a mean ratio that is relatively constant 
within a selected group of persons, or for a given 
aggregate of property valuations. We have only to re- 
call the use of statistics in education, in social institu- 
tions such as hospitals, asylums, prisons, etc., to 
recognize the wide field within which this method is 
useful. 

Statistics are of indispensable service to the legis- 
lator, to the economist, and to the student of social 
problems. But for this method, uniformities in phe- 
nomena so comphcated and seemingly without law, 
and whose causes are so obscure, could never have 
been discovered. But practical utiHty is not the only 



206 ELEMENTARY LOGIC 

value which the statistical method may claim. Sta- 
tistics afford valuable data for science; they suggest 
new lines of inquiry, and set new problems, the solution 
of which enlarges the boundaries of science. 

It was observed, in discussing the calculation of 
chances, that the striking deviations from the results 
deduced in accordance with the theory suggest new 
hypotheses; and these, verified, add new territory 
to the domain of science. So with statistics; any 
marked deviation from the computed average or ratio 
hitherto maintained constitutes a fresh problem; and 
the methods of observation, experiment, and hypothesis 
are brought into use, with the not infrequent result of 
new laws being discovered, and causes hitherto hidden 
being brought to light. 

Statistics thus give opportunity for sagacious sugges- 
tions and fruitful investigations. They are of great 
aid also in rendering observation more varied and 
more precise; they give opportunities for experiment, 
— nay, they are of the nature of experiments ; and, 
finally, they are serviceable in testing hypotheses, 
by making verification more critical and more com- 
plete. I have made use of the investigations of 
Weather Forecaster Dunn in illustrating the method 
of concomitant variations, but these investigations 
afford so good an example of the scientific use of 
statistics, that I shall borrow from them again. Select- 
ing the period from March 22 to May 16 in the 
year 1891 (the time of the grippe epidemic in New York 
City), Mr. Dunn prepared a chart which gave the total 



THE THIRD SPECIAL PROBLEM 207 

number of deaths from grippe in this period. Other 
statistics were gathered relating to the weather condi- 
tions, when there was the greatest number and the 
least number of cases, and when the fatality was the 
greatest. Now, it was by means of such statistics that 
Mr. Dunn succeeded in estabHshing, with very great 
probabihty, a causal connection between weather con- 
ditions — particularly degree of humidity and tem- 
perature changes — and the increase and diminution 
of this disease. 



CHAPTER XIV 

GENERALIZATION FROM EXPERIENCE AND 
ANALOGY 

Section 36 
inductive generalization and its varieties 

This seems to be an appropriate place to discuss the 
function and logical value of two closely allied forms 
of inference we have already explained in an earlier 
chapter. 

I. Inductive inference, it was there explained, pre- 
sents two varieties — one of which I will venture to 
call inductive generalization; the other is known as 
analogical inference, or analogy. Inductive generahza- 
tion must not be confounded with the hypothetical step 
in explanation; to frame an hypothesis and to draw 
an inference are distinct things. Inductive generaliza- 
tion is the extension of what has been observed in cer- 
tain cases to other cases which have not been observed; 
an hypothesis involves a conception of something which 
may be very unlike what has been observed. Induc- 
tive generalization is not explanation of anything in 
present experience; hypothesis, as we have seen, is a 

208 



EXPERIENCE AND ANALOGY 200 

method of explanation, — it is framed for the purpose 
of explaining what is given in experience. Inductive 
generalization has tw^o functions : — 

(i) It constitutes a prehminary step to scientific ex- 
planation ; 

(2) It establishes other uniformities or approximate 
uniformities of experience, for which no explanation is 
yet found, and which may be destined to remain merely 
empirical laws. 

Generalizations from experience, according to Mr. 
Mill, are of two sorts: absolute generahzations, and 
those which are approximate only. Absolute gener- 
alizations are derived from experience that has pre- 
sented no exceptions to the observed uniformity which 
is the foundation of the inference; approximate gen- 
erahzation is based upon an experience in which, 
in the great majority of instances, the same thing 
has occurred, but in which exceptions to this rule 
have also occurred. Absolute generahzations are 
expressed in universal propositions ; approximate 
generahzations, in particular propositions. Thus, all 
crows are black, is an absolute generalization; most 
men seek their own interests first, is an approximate 
generahzation. 

Approximate generalizations are of little scientific 
value, but they may possess great value for conduct; 
our conduct in matters of great importance sometimes 
has no other guidance. This is especially the case with 
much of our conduct in relation to our fellow-beings. 
Mr. Mill observes, "All propositions which can be 



210 ELEMENTARY LOGIC 

framed respecting the actions of human beings, are 
merely approximate; we can (for example) only say 
that most persons of a particular age, profession, 
country, rank in society, have such and such quaHties; 
most persons when placed in such and such circum- 
stances act in such and such a way" ("Logic," p. 418). 

2. Analogical Inference. — The service which ana- 
logical inference renders to science has been pointed 
out in the discussion of hypotheses; it remains to con- 
sider the other functions of this mode of inference, and 
to estimate its logical value. The scientific value of 
analogy, apart from its suggestiveness in explanation 
by hypothesis, is slight. This inference is relatively 
strong only when the resembling properties on which 
it is based are not merely numerous, but are important ; 
and the difficulty of estimating the character of the re- 
sembling properties, instead of merely counting them, 
renders the use of analogy of doubtful service. But, 
shght as is the service of this inference to science, its 
practical value is often great. 

For many of our strongest beliefs and those which 
profoundly influence our actions, analogy is the only 
evidence we go upon. For example, the conviction we 
have, that our fellow-beings are men of like passions 
as ourselves, suffer and enjoy what we do, and recog- 
nize the obhgations we accept, etc., has no other logi- 
cal ground than the inference from analogy. The 
expressive acts and deeds of our social fellows are 
the basis on which we attribute to them the same 
thoughts, feehngs, and purposes that we express by like 



EXPERIENCE AND ANALOGY 211 

acts and deeds. So is it with our belief in purpose, 
design, both in the case of productions we attribute to 
beings hke ourselves, and in the case of organic struc- 
tures in nature. The principle of teleological explana- 
tion is analogy. 

The fact that analogy is so common a method of 
forming behefs, and the fact that for so many of them 
there is no other justification, make desirable a more 
expHcit statement of the principles which should 
regulate our use of analogy. These regulative prin- 
ciples are the following : — 

(i) The importance of the resembhng properties, 
rather than their number, should determine our behef. 
Weigh, rather than count, the points of likeness and 
difference. 

(2) The strength of an inference from analogy is 
proportioned to the reason we have for beheving any 
property of that from which the inference starts is 
connected with that property to which the inference 
proceeds; a single property which we have reason to 
believe is thus connected with the conclusion should 
have more weight in determining our behef, than nu- 
merous properties that afford no evidence of such 
connection. 

(3) In cases in which our inference is based upon the 
number of agreeing circumstances relative to the number 
of disagreeing circumstances, it is important that all 
points, both of agreement and difference, should be 
under view. 



CHAPTER XV 

FALLACIES 

Section 37 

fallacies incident to reasonings upon matters 

OF FACT 

In this chapter we shall describe and explain those 
fallacies to which we are hable in the processes of reason- 
ing which we have already explained. For convenience 
we divide fallacies into four groups, three of which 
correspond to the three special problems of science. 

Accordingly, the first group includes the errors in- 
cident to the ascertainment of causal connection by 
observation and experiment. The second group in- 
cludes errors of explanation by hypothesis; and the 
third group, the fallacies incident to calculation of 
chances and the statistical method. Into the last 
group we put the fallacies of generalization and false 
analogies. 

Fallacies in Explanation by Observation and Ex- 
periment. — A part of the fallacies in our first group are 
commonly fallacies of observation ; but, properly speak- 



FALLACIES 213 

ing, observation is not fallacious ; only inference, or dis- 
cursive thinking, can be so characterized. Observation 
can be defective and otherwise faulty, and, consequently, 
be a psychological source of fallacies, as are our passions, 
our subjective biases of various sorts; but fallacy is 
committed only when we accept something as evidence 
for a behef which is either not evidence at all, or is 
inadequate evidence. Owing to defective observa- 
tion, I may wrongly infer that there is a causal connec- 
tion between two phenomena, A and B, just as passion 
or prejudice may cause me to give assent to a proposi- 
tion for which there are no vahd reasons; the fallacy 
in so doing consists in mistaking evidence. 

This logical error must be distinguished from the 
psychological causes of it, which in these instances were 
wrong perception, prejudice, and passion. The common 
feature of all the fallacies of this first group is behef 
in causal connection in the absence of evidence, or upon 
insufficient evidence, the data not warranting the in- 
ference. Turning first to observation, as one of the 
fruitful sources of these fallacies, we note there are here 
two causes to which these fallacies are due: (i) over- 
sight of material circumstances, — non-observation ; 
(2) confusion of perception with inference. 

(i) An illustration of errors arising from oversight 
of material circumstances is afforded by advertisements 
of patent medicines, accounts of cures effected by 
them ; only the successes being noted, the failures not 
observed. Another illustration is the belief that 
Friday is an unlucky day, only mishaps and fatalities 



214 ELEMENTARY LOGIC 

being observed, the fortunate incidents being over- 
looked. 

What predisposes the mind to this fault in observa- 
tion is the circumstance that coincidences impress us, 
and strongly so, if they are for any reason deeply inter- 
esting to us; that interest may be painful no less than 
pleasant; while coincidences on the other hand pass 
unnoticed. Carelessness, lack of interest, and conse- 
quently inattention to material circumstances are of 
course also causes of this mal-observation. 

(2) The errors which arise from confusion of 
perception with experience are common enough, and 
almost unavoidable by untrained minds, without the 
corrections of experiment, or of careful comparison 
with the observations of others. Mr. Mill gives the 
following example of this form of wrong observation: 
"People fancied they saw the sun rise and set; and 
stars revolve in circles around the pole; and while 
they did so, they stubbornly refused to accept the Co- 
pernican theory." Inference here was mistaken for 
perception. People do not see the sun rise, etc. ; they 
infer this from what they do see. 

But, even were our observation faultless, and as 
complete as our faculties employed in the most care- 
ful manner could make it, even then the inference 
to causal connection would not be secure from fal- 
lacies. So great is the complexity of natural phe- 
nomena, so manifold and intricate the threads of 
connection between them, that no observation, how- 
ever careful and however supplemented by experi- 



FALLACIES 21$ 

ment, can disentangle this web and eliminate com- 
pletely the non- causal circumstances in the case of 
any given phenomenon. 

We have seen that the five methods of observation 
and experiment are the instruments by the use of 
which we attempt this resolution of phenomena and 
the discovery of causal connections; we have pointed 
out the hmitations and defects of these methods; and 
we have seen what are the reasons why we can 
reach by them conclusions that are only probable, 
this probabihty in many cases being of a low 
degree. Hence, the fallacy to which we are exposed 
in relying upon these methods is that of overesti- 
mating the evidence they afford of causal connection. 

Among these methods the one which most exposes 
us to this fallacy is the method of single agreement. 
The fallacy of post hoc propter hoc springs especially 
from the use of this method; and, inasmuch as it is 
this method that, from the nature of the case, is most 
commonly employed in reasoning upon political affairs, 
this fallacy is the almost universal sin of untrained 
reasoners, especially partisans and doctrinaires who 
dabble in sociological and economic matters. Exami- 
nation of very many popular beliefs, those of which we 
are most tenacious, would show that they have no 
better ground than this customary conjunction in ex- 
perience from which we infer a causal connection; 
they rest on the fallacy of post hoc ergo propter hoc. 

Nor is the method of difference — the surest and 
most reliable of all the methods — exempt from this 



2l6 ELEMENTARY LOGIC 

liability to false inference. Suppose I am experiment- 
ing with the use of a particular kind of food in 
following this method; I select for comparison two 
instances, the instance in which I take the food and 
am made ill, and the instance in which I do not take 
the food and remain well. Now, could I be certain that 
my condition of health in the second instance was the 
same as in the first instance save in the one circum- 
stance, that food was not taken in the one instance, and 
was taken in the other instance, then I should be justi- 
fied in being certain of a causal connection between 
taking that food and being made ill. 

But there is one unavoidable source of possible 
error in all such experiments: I must in this case 
compare two instances that are separated by some 
interval of time; and I can by no means be sure that 
in that interval a change has not taken place in my 
general condition, that was not a predisposing factor 
at least, in the illness which resulted from taking 
the food. 

One other circumstance about these methods we are 
liable to overlook is their hypothetical and ideal char- 
acter. As we have seen, they presuppose conditions to 
which we can only approximate in the actual situations 
which we most employ in them. Scientific investigators 
know these limitations of the methods they employ; 
and do not fall into the error of overestimating the 
evidence they afford; but minds not thus disciplined 
are apt to measure the strength of the evidence by the 
canons; and to overlook the fact that these canons 



FALLACIES 2 1 / 

can be only partially complied with in our actual 
world. 

Fallacies in Explanation by Hypothesis. — The fal- 
lacies of this group have their source in the processes 
of verification; and they arise either from an incom- 
plete deduction and consequently incomplete compari- 
son with facts, or from overlooking the difference be- 
tween verification and complete proof. If the deduction 
from an hypothesis does not extend beyond the particular 
phenomenon it is constructed to explain, the evidence 
for the truth of that hypothesis is shght; oversight of 
this very hmited test of the hypothesis leads to over- 
estimation of the evidence in support of it. Even in 
case the deduction is extended to other phenomena, 
if those phenomena are of rather limited range and 
of simple character, we are hable to overestimate the 
evidence afforded by this fuller verification. 

The disposition to be satisfied with incomplete veri- 
fication has its source in our various subjective interests, 
propensions, and biases of mind ; especially is this the 
case with social, economic, ethical, and rehgious hypoth- 
eses. If they accord with our habits, our inchnations, 
and favorite ways of viewing things, we are little dis- 
posed to look for facts that disprove them. The strength 
of our behef in them is not according to the objective 
reasons that exist, but according to their appeal to our 
subjective interests and appreciations. Most of these 
beliefs are extra-logical ; and the fallacy we commit in 
them consists in oversight of the distinction between 
logical grounds and psychological motives or causes. 



2l8 ELEMENTARY LOGIC 

The other fallacy in verification has its source in 
mistaking complete verification for complete proof. 
It is substantially the fallacy of affirming the consequent 
in the hypothetical syllogism. To draw the conclusion 
that an hypothesis is certainly true because the given 
phenomena are just what they would be if this hypothe- 
sis were true, overlooks the possibility of what Mill 
calls plurality of causes. 

There is one more error, somewhat the converse of 
this fallacy, which arises from overlooking the difference 
between disproof and incomplete verification. Ob- 
jectors to a given hypothesis are prone to this fallacy 
of inferring that an hypothesis is false if it fails to 
explain all the phenomena. Critics of Darwin drew 
such an inference from his frank admission that there 
were outstanding facts which his theory did not satis- 
factorily explain; and some of these objectors were 
confident that Darwin had logically abandoned his 
hypothesis, when he admitted, that if there was a single 
fact that could not be harmonized with his hypothesis, 
he must abandon it. The unexplained facts were at 
once pointed out, with the triumphant inference that 
Darwin's theory was overthrown by his own admis- 
sion. Now, the fallacy which these logical refuters 
of Darwin committed, is the fallacy of confounding 
facts not explained by an hypothesis with facts that 
contradict that hypothesis. 

Fallacies Incident to the Calculation of Chances 
and the Method of Statistics. — The theory of chance, 
we have seen, assumes that the causes which determine 



FALLACIES 219 

the particular event are so adjusted to each other in 
their influence, that, in a sufficiently long run, as many- 
events of one particular sort will occur as of any other 
sort of possible events; any one side of the die will 
come up as many times as any one of the other five 
sides. Consequently, when, in a limited succession 
of instances, the same event occurs a disproportionate 
number of times, the inference seems to be justified 
that some particular cause is operative. In some cases 
it has been shown that this inference is incorrect ; and 
the error is a consequence of overlooking the tendency 
of such phenomena to a sort of rhythmic recurrence of 
the same succession; that is, a run of events of the same 
character — six-spots — is followed by a run of those 
of different character, — say aces, — and a return of 
the run to sixes, etc. 

It should not, however, be maintained that this fal- 
lacy of inferring a special cause is committed in all 
cases where a like excess of actual over theoretical 
coincidences occurs. In the phenomena of telepathy, 
or alleged thought transference, it is very confidently 
maintained that the coincidences are so much and so 
persistently in excess of mere chance coincidences, that 
no other conclusion is reasonable but the existence of 
a particular cause. It does not, however, come within 
the province of logic to determine the truth or error 
of such inferences. If they should be proved to be 
erroneous, . the fallacy would be the result of mistak- 
ing recurring non- causal coincidences for causally con- 
nected ones; and the source of this error would be 



220 ELEMENTARY LOGIC 

an oversight of the tendency of such phenomena to 
form habits, — of the same sort as are shown in dice 
throwing, selecting cards, balls from a box, etc. 

The method of statistics is liable to two special 
fallacies: one of these is that of inferring a causal 
determination of a particular phenomenon from uni- 
formity which holds true of a group to which this par- 
ticular phenomenon belongs; thus, from the fact that 
in a certain city the rate of mortality is j^^, it is inferred 
that A, who lives in that city, has ninety-nine chances 
of living against one of dying in a specified period of time. 

This fallacy is akin to the fallacy of assuming 
that whatever can be affirmed of a class can also be 
affirmed of every individual of that class ; but the exact 
cause of this fallacy is an erroneous conception of the 
statistical method, and what that method assumes; 
and the fallacy really consists in confounding the statis- 
tical method with the calculation of chances. The 
other fallacious use sometimes made of statistics is 
the attempt to prove by means of them that human 
actions have natural causes, as do all phenomena in 
nature; in other words, the attempt is made to estab- 
lish a doctrine of determinism by the aid of this method. 
Human actions, so runs the argument, cannot be free, 
because it is ascertained that in any specified group of 
persons, and for a given period of time, a relatively 
constant number of actions of a particular sort, say 
murder, are performed ; there must be some constantly 
acting cause, and such a cause is incompatible with 
free will. 



FALLACIES 221 

Now, such an argument involves a twofold miscon- 
ception : a misconception of the method of statistics, 
and a misconception of the nature of free action and 
the conditions of it. The latter misconception is a 
metaphysical one, and the exposure of it does not 
fall to the logician. The first error overlooks the fact 
that the method of statistics does not consider indi- 
viduals as individuals in the averages and constant 
ratios it establishes. It is entirely erroneous to infer 
that, since causes determine this constant ratio, they 
do so by determining the actions of all the individuals 
who composed the group. There is room in such 
aggregates as the method of statistics contemplates, 
for all the individual freedom of choice that the stoutest 
champion of this doctrine need contend for. 

Fallacies in Generalization and in Analogy. — The 
fallacy to which inductive generahzation is prone is 
that of generalizing from too narrow a datum. In- 
stances of such unjustified inferences abound in popu- 
lar beHefs. Critical examination would surprise most 
people by the discovery that very many of the beliefs 
they think are securely based, have really no other 
foundation than an experience of narrow range. In 
many cases it is an experience of few instances on which 
the broad generalization rests, especially if those in- 
stances are such as excite or appeal strongly to our 
predilections and our prejudices, our Hkes and our 
aversions. 

The absence of contradictory instances greatly helps 
this tendency to hasty generahzation. But perhaps the 



222 ELEMENTARY LOGIC 

strongest factor in inclining our minds to premature 
generalization is a disposition, a structural principle of 
our minds, and rational in itself, — the disposition 
to look for uniformity, the prejudice in favor of 
order, the propensity to see in all cases not yet under 
observation the same nature that we find in those 
we know. We want uniformity; we want the future 
to be like the present; we are impatient of delay; 
we dislike to wait for contradictions of what seems 
to be the uniformity we have already begun to accept, 
and especially if it be a uniformity that is congenial 
to us. 

A practical interest asserts itself in these generaliza- 
tions and tends to make them hasty; it is the need 
for action, for adjustment to coming situations, espe- 
cially to the behavior of our social fellows. If what 
I have found to be true of some men I can now beheve 
to be true of men I have yet to know personally, I 
can the more successfully plan my future actions in 
reference to them; knowing what I would have to ex- 
pect from them in any situation, I can prepare to 
meet that situation. Thus we hurry our generaliza- 
tions, because we have practical interests that we 
think will not wait the slow testing of experience. 

Thus, a variety of motives and interests, partly 
emotional, partly intellectual, and partly practical, con- 
spires to this error of generalization from insufficient 
data, conspires to produce this illusion of the logical 
faculty, leading it to overestimate the evidence for its 
conclusion. 



FALLACIES 223 

Our exposition of analogical reasoning has made it 
quite obvious that the one fallacy to which it is exposed 
consists in counting resemblances or differences instead 
of weighing them. The unpracticed reasoner is prone 
to suppose the inference is necessarily strong in pro- 
portion to the excess of the points of Ukeness over the 
points of difference; and he is hable to overlook the 
importance either of the resembhng properties or of 
the properties that present differences. Reasoning 
upon social and poUtical matters is a fruitful field 
for false analogies. No argument is more specious, 
and no argument so readily captivates the imagination 
and is so successful with the ordinary mind. 



APPENDIX 



PRACTICAL EXERCISES AND QUESTIONS 

CONCEPTS, NAMES, TERMS 

1. Explain the following distinctions: simple, com- 
plex, universal, singular, collective, concrete, abstract, 
positive, negative, absolute, relative. 

2. In the following, which concepts are universal, 
which are singular, and which are collective? man, 
army, stone, science, congress, Edward VII, charity, 
union, Venus, the Pope, courage, the year 1905. 

3. State whether the following concepts in the in- 
stances given are concrete or abstract : quality, mercy, 
justice, meekness, truth, righteousness, blue, humanity, 
nation, nationality, equahty. 

(i) The quality of mercy is not strained. 

(2) Mercy rejoiceth against judgment. 

(3) Meekness is a virtue in a strong man. 

(4) Truth crushed to earth shall rise again. 

(5) Righteousness exalteth a nation. 

(6) The sky is blue. 

(7) Humanity is destined to finer development. 

(8) He is of Greek nationahty. 

224 



PRACTICAL EXERCISES AND QUESTIONS 225 

4. Give the logical characteristics of the following 
(by logical characteristics are meant the distinctions, 
complex, simple, universal, singular, collective, etc.) : — 

man charity parhament 

conscience sweetness king 

wood Pole father 

virtue the sun genius 

peer feehng army 

happiness milHon John Smith 

5. How do you explain extension, intension, connota- 
tion, and denotation of a name or concept ? 

6. Arrange the following with reference to extension 
and intension, placing them in the order of diminishing 
extension: man, animal, ruler, living being, emperor, 
king of England, quadruped, horse, racing horse, 
body, star, Sirius, heavenly body. 

7. Explain division, classification, and definition, 
and the relation between division and definition. 

8. Divide the following, carrying the division three 
steps in each instance : trees, books, buildings, govern- 
ment, sciences, metals. 

9. Explain genus, species, difference, property, and 
accident, and give an example of each. 

ID. What distinction can be made between accident 
and property, and between proximate species and lowest 
species ? 

11. Give definitions of the following: circle, history, 
quadruped, poetry, money, government, the North Pole. 

1 2 . What are the faults in the following definitions : — 
(i) Man is a two-legged animal without feathers. 

Q 



226 ELEMENTARY LOGIC 

(2) A triangle is a plane figure having three sides, 

three angles, and the sum of the angles is 
equal to two right angles. 

(3) A king is an hereditary ruler of a kingdom 

having an extensive area and a dense popu- 
lation. 

(4) Vice is the opposite of virtue. 

(5) A gentleman is a man having no visible 

means of subsistence. 
(7) Man is a self-knowing animal. 
13. Examine the following divisions and state why 
they are faulty : — 

(i) Rectilinear figures are divided into triangles, 
parallelograms, rectangles, and polygons 
of more than four sides. 

(2) Wars are divided into civil and destructive. 

(3) Allegiance is either natural and perpetual 

or local and temporary. 

JUDGMENTS AND PROPOSITIONS 

1. Distinguish a judgment from a grammatical sen- 
tence. 

2. In what different ways can a judgment be ex- 
pressed ? 

3. Explain the hypothetical and the disjunctive judg- 
ments. 

4. Analyze the following sentences. (The logical 
analysis of a sentence is reduction of it when necessary 
to the proposition or propositions it contains, the propo- 
sitions being made as simple as possible so as clearly to 



PRACTICAL EXERCISES AND QUESTIONS 22/ 

express the relation between the subject and predicate 
terms. Thus, the sentence, "They have rights who dare 
maintain them," analyzed gives, "Those who dare main- 
tain their rights are those who have rights"; and the 
sentence, "Where there's a will there's a way," gives, 
"The situations in which there is a will are the situa- 
tions in which there is a way," or "To have a will is to 
find a way." The two relations between subject and 
predicate terms to which propositions can be reduced 
are the relation of attribute to subject or of class to 
class; and in the analysis of sentences it is best to re- 
duce them to propositions which assert one or the other 
of these relations.) 

(i) Then to side with truth 'tis noble, when we 
share her wretched crust. 

'2) Truth crushed to earth shall rise again. 

(3) Honor and shame from no condition rise. 

(4) Some murmur when their sky is clear. 
^5) Castles of the great are jails. 
(6) Stone walls do not a prison make, nor iron 

bars a cage. 
'7) 'Tis equal peril to go or to remain. 

(8) He who is capable of making a pun is capable 
of picking a pocket. 

(9) New occasions teach new duties. 
'10) They must ever up and onward who would 

keep abreast of truth. 
'11) All's well that ends well. 
'12) All cannot receive this saying. 
'13) A friend should bear a friend's infirmities. 



228 ELEMENTARY LOGIC 

(14) The path of glory leads but to the grave. 

(15) Few shall part where many meet. 

(16) One ruddy drop of manly blood the surging 

sea outweighs. 

(17) A little consideration of what takes place 

around us every day would show us that 
a higher law than our wills regulates 
events. 

(18) There is less intention in history than we 

ascribe to it. 

(19) What we do not call education is more pre- 

cious than what we call so. 

(20) Laurel crowns cleave to deserts, and power 

to him who power exerts. 
5. Distinguish the kinds of judgments and the kinds 
of propositions in the following : — 
(i) Whatever is, is right. 

(2) Planets revolve from east to west. 

(3) If a body is heated, it expands. 

(4) All exercises are not so easy as they look. 

(5) Honor lost, all is lost. 

(6) Where there's a will there's a way, 

(7) Plane figures are either rectilinear or curvi- 

Hnear. 

(8) What is not an element is not a metal. 

(9) Life's not all beer and skittles. 

(10) How vain that second hfe in other's breath! 

(11) We must either control our passions or be 

their slaves. 

(12) He can't be wrong whose life is in the right. 



PRACTICAL EXERCISES AND QUESTIONS 229 

(13) Only the actions of the just smell sweet, and 

blossom in the dust. 

(14) Good men sometimes do wrong. 

(15) Some men are always in the wrong. 

(16) Few shall part where many meet. 

(17) Only honorable actions deserve praise. 

(18) Fame is no plant that grows on mortal soil. 

(19) Some of those present were all who engaged 

in the sport. 

(20) Only a few did not escape. 

6. Explain the negative judgment, and illustrate by 
an example the positive value of negation. 

7. What do negative propositions assert? 

8. What is the quality of the following propositions : 
(i) Few and short were the prayers we said. 

(2) Few shall part where many meet. 

(3) What is not an element is not metal. 

(4) Only members vote. 

9. Explain quantification. Show how it applies to 
each term in the proposition. 

ID. By using the symbols A, E, I, and O, indicate 
the quantity and quahty of the propositions given under 
question 5. 

II. Quantify the following propositions. 

(In quantifying reduce each proposition to one of the 
formulae explained under the topic Quantification. Also 
make use of the circles to illustrate quantification.) 

(1) A and B are always together. 

(2) Some murmur when their sky is clear. 

(3) All lawyers are not knaves. 



230 ELEMENTARY LOGIC 

4) All the perfumes of Arabia will not sweeten 
this little hand. 

5) None know where the shoe pinches but the 
wearer, 

6) He is gentle that doeth gentle deeds. 

7) Most men prefer their own interests. 

8) Not every one who saith to me Lord, Lord, 
shall enter the kingdom. 

9) To be great is to be misunderstood. 

10) Only a few were saved. 

11) Some of the voters were the only ones who 
voted. 

12) A fool at forty is a fool indeed. 

13) No one is always happy. 

14) A stitch in time saves nine. 

15) Not many wise men are called. 

16) Few are chosen. 

17) We spoke not a word of sorrow. 

18) There is none good but one. 

19) Uneasy lies the head that wears the crown. 

20) The North Pole is not yet reached. 

IMMEDIATE INFERENCE 

1. Explain and illustrate equipollence and opposi- 
tion. 

2. Name the different logical relations under equi- 
pollence and under opposition. 

3. Arrange the following propositions in pairs so 
that a logical relation shall be shown in each pair of 
propositions, and state what this relation is : — 



PRACTICAL EXERCISES AND QUESTIONS 23 1 

(i) All material substances possess gravity. 

(2) No material substances do not possess gravity. 

(3) What does not possess gravity is not a material 

substance. 

(4) No material substances possess gravity. 

(5) Some material substances do not possess grav- 

ity. 

(6) Some things which possess gravity are mate- 

rial substances. 

(7) Some things not possessing gravity are mate- 

rial substances. 

(8) No substances possessing gravity are material 

substances. 
Obvert the following propositions : — 
(i) All metals are useful. 
(2) All organic substances contain carbon. 
(2) All mammahans are vertebrates. 

(4) Whatever is necessary exists. 

(5) No men are always happy. 

(6) Nothing great is easy. 

(7) Some mistakes are culpable. 

(8) Few escape misfortune. 

(9) Some mistakes are not culpable. 

(10) Only members vote. 
Convert the following : — 

(i) A stitch in time saves nine. 

(2) Few shall part where many meet. 

(3) A man's a man for a' that and a' that. 

(4) Uneasy lies the head that wears the crown. 

(5) Every little makes a mickle. 



232 ELEMENTARY LOGIC 

6) Few men are perfectly content. 

7) No men are absolutely bad. 

8) Where there's a will, there's a way. 

9) All's well that ends well. 

10) ^ and B are always together. 
6. State the contrapositive to the following : — 

i) All living tissue is organic. 

2) No man loves dishonor. 

3) Some promises are better broken than kept. 

4) Knowledge is power. 

5) Some mistakes are not culpable. 
Contradict and give the contrary to the follow- 



. 7. 
ing: 



i) Whatever is, is right. 

2) All the world is gone after him. 

3) No wrong can remain unrighted. 

4) Some men are always in the wrong. 

5) No man is a hero to his valet. 

Give every opposition possible to the following : — 
i) Life is not all beer and skittles. 

2) Some just acts are inexpedient. 

3) All's well that ends well. 

4) The longest road has a turn. 

5) Only the brave deserve the fair. 

6) Whatever is, is right. 

7) Some good actions are not rewarded. 

8) None but brave men dare always do right. 

9) All roads lead to Rome. 

10) Men of fair promises are often not to be 

trusted. 



PRACTICAL EXERCISES AND QUESTIONS 233 

9. From each of the following propositions, as a prem- 
ise, pass to as many propositions as are logically re- 
lated to the given proposition : — 

(i) Perfect happiness is impossible. 

(2) Few are acquainted with themselves. 

(3) Talents are often misused. 

(4) No knowledge is useless. 

(5) He jests at scars who never felt a wound. 

(6) Familiarity breeds contempt. 

(7) All's well that ends well. 

(8) Some stars are not seen. 

(9) Nothing is worth doing at all that is not worth 

doing well. 

(10) Every little makes a mickle. 

MEDIATE INFERENCE. THE SYLLOGISM 

1. Construct a categorical, a hypothetical, and a dis- 
junctive syllogism respectively. 

2. Explain major, minor, and middle terms. 

3. Define figures of the syllogism and construct a 
syllogism in each of them. 

4. Explain moods and the maximum number of 
them. 

5. In the categorical syllogism, prove the following 
rules : — 

(i) One premise at least must be universal. 

(2) One premise at least must be afiirmative. 

(3) In Fig. I the minor must be affirmative. 

(4) In Fig. II one premise must be negative. 

(5) In Fig. I the major premise must be universal. 



234 ELEMENTARY LOGIC 

6. Construct a dilemma, a sorites, and an epichi- 
rema, and distinguish between constructive and de- 
structive, and between complex and simple dilemmas, 
also between the Aristotelian and the Goclenian 
sorites. 

7. Expand the following enthymemes into complete 
syllogisms : — 

(i) Blessed are the meek, for they shall inherit 
the earth. 

(2) Some pleasures are not praiseworthy, hence 

some pleasures are not honorable. 

(3) A nation may depose a bad king ; for it has a 

right to good government. 

(4) Law is an abridgment of liberty, and conse- 

quently of happiness. 

(5) If he did not steal the goods, why did he con- 

ceal them as no thief ever fails to do ? 

(6) He cannot be a gentleman, for no gentleman 

would do such a thing. 

(7) A body cannot move; for to do so it must 

move where it is or where it is not and it 
can do neither of those things. 

(8) If it is fated that you are to recover you will 

do so whether you call in a doctor or not, 
and if it is fated that you will not recover, it 
is useless to call in a doctor ; consequently, 
it is useless to call in a doctor. 

(9) Our ideas reach no farther than our experi- 

ence; we have no experience of obvious 
attributes. 



PRACTICAL EXERCISES AND QUESTIONS 235 

(10) I infer that some stupid persons must have 

passed in the last examination. 

(11) Discontent is an essential condition of prog- 

ress, — but discontent means sorrow. 

(12) That which causes a balance of good is right ; 

therefore, persecution may sometimes be 
right. 

(13) How can one maintain that the insane should 

never be punished who maintains that 
they should always be benefited ? 

(14) No man should fear death; for it is accord- 

ing to nature. 

(15) All human things are subject to decay; and 

when fate summons, monarchs must obey. 

(16) He is free who hves as he wishes; the bad, 

therefore, are not free. 

(17) Blessed are the peacemakers, for they shall 

be called the children of God. 

(18) There can be no religion without infallibility ; 

for no reUgion is possible without a visible 
church, no church without government, no 
government without sovereignty, and no 
sovereignty without infallibility. 

(19) All love happiness; all love Hfe. 

What premises have the following proposi- 



tions : 



(i) He is not a wise man. 
(2) Knavery and folly sometimes go together. 
. (3) Some victories are won by accident. 
(4) He must die. 



236 ELEMENTARY LOGIC 

(5) A college education is not always necessary 

to success. 

(6) The meek must be blessed. 

9. Draw conclusions from the following premises 
where a conclusion is admissible, and give reasons why 
some of the premises do not give conclusions. 

(i) No birds are biparous; all feathered animals 
are birds. 

(2) Sodium is a metal; sodium is a very dense 

substance. 

(3) Violations of law should be punished; lying 

is not a violation of law. 

(4) Most men prefer their own interests; A is a. 

man. 

(5) All men are mortal; no men are perfect. 

(6) Scarlet fever patients have high temperature ; 

X has a high temperature ; X has not a high 
temperature. 

(7) Only express trains do not stop at this station; 

the last train was an express train. 

(8) If it rains, he will not come ; he did not come. 

(9) If it rains, he will not come ; it did not rain. 

(10) If^is5, Cis-D; and if E is jP, G is fi" ; Cis 

not D and G is not H. 

LOGICAL FALLACIES 

I. Explain the fallacies in the following, giving the 
technical names for them : — 

(i) All who think this man innocent think he 
should not be punished; you think he 



PRACTICAL EXERCISES AND QUESTIONS 237 

should not be punished ; therefore you must 
think he is innocent. 

(2) Whatever is vicious should be punished; in- 

temperance is not vicious and therefore 
should not be punished. 

(3) He must have stolen the goods, for he con- 

cealed them, which every thief does. 

(4) Everything permitted by law is morally right ; 

and therefore whatever is morally right is 
permitted by law. 

(5) Express trains only do not stop at this station; 

and as the last train did stop, I infer it was 
not an express train. 

(6) Over credulous persons ought never to be 

believed : and since some ancient historians 
are untrustworthy, they must have been over 
credulous. 

(7) Nearly all the satellites revolve from west to 

east ; the moon therefore must revolve from 
west to east. 

(8) Had Pitt carried out the doctrine of free frade, 

he would have been a great statesman ; but 
he did not carry out that doctrine; there- 
fore he was not a great statesman. 

MATERIAL FALLACIES 

I. Describe the following fallacies: accident fallacy, 
fallacy of composition, and the converse fallacy of 
division. 



238 ELEMENTARY LOGIC 

2. How distinguish between the fallacy of ambiguity 
and accident fallacy? 

3. What is fallacy a dido simpliciter ad dictum se- 
cundum quid? What is the converse fallacy and how 
are these fallacies related to the accident fallacy ? 

4. Distinguish between petitio principii and ignora- 
tio elenchi. 

5. In what two ways is petitio principii committed, 
and what forms of ignoratio elenchi are there ? 

6. Describe and name the fallacies in the following 
arguments : — 

(i) Slavery is a natural institution, and therefore 
it ought not to be abolished. 

(2) We know that God exists, because the Bible 

tells us so; and we know that whatever 
the Bible affirms is true, because it is of 
divine origin, 

(3) Nations are justified in revolting when badly 

governed, because every nation has a right 
to good government. 

(4) Some holder of a ticket is sure to draw the 

prize; and, as I am a ticket holder, I am 
sure to draw the prize. 

(5) What fallacy did Columbus commit, when he 

made the egg stand on end by breaking in 
the end of the egg? 

(6) What fallacy was the humorist afraid of when 

he said he would not accept a demonstra- 
tion in mathematics, until he knew what use 
was to be made of it ? 



PRACTICAL EXERCISES AND QUESTIONS 239 

(7) Improbable events happen every day; now, 

what happens every day is a probable event ; 
therefore, improbable events are probable 
events. 

(8) A miracle is incredible because it contradicts 

the laws of nature. 

(9) Every hen comes from an egg, every egg comes 

from a hen ; therefore, every egg comes from 
an egg. 

(10) The Germans are beer drinkers; Hans, be- 

ing a German, must also be a beer drinker. 

(11) What we eat grew in the fields; loaves of 

bread are what we eat; therefore, loaves 
of bread grew in the fields. 

(12) Wine is a stimulant; therefore, in every 

case where a stimulant is harmful, wine 
is harmful. 

(13) Gold and silver are the wealth of a country; 

consequently the diminution of gold and 
silver by exportation must be the diminu- 
tion of the wealth of a country. 

(14) If I am to pass this examination, I shall pass 

it whether I answer correctly or not ; if I 
am not to pass it, I shall fail whether I 
answer correctly or not; therefore, it is 
of no consequence how I answer the ques- 
tions. 

(15) All the trees in the park make a dense shade ; 

this oak is a tree in the park, and conse- 
quently it makes a dense shade. 



240 ELEMENTARY LOGIC 

(i6) Whoever intentionally kills another should 
suffer death; a soldier should there- 
fore suffer death, since he intentionally 
kills. 

(17) Every rule has exceptions; this statement 

is a rule, and therefore has exceptions; 
therefore, there are some rules that have 
no exceptions. 

(18) Repentance is a good quality; wicked men 

abound in repentance; and therefore 
they abound in what is good. 

(19) Meat and drink are the necessaries of life; 

the revenues of the king were spent on 
meat and drink, and consequently they 
were spent on the necessaries of hfe. 

(20) We charged him (King Charles the Second) 

with having broken his coronation oath, 
and we are told that he kept his marriage 
vows; we accuse him of having given up 
his people to the merciless infliction of the 
most hot-headed and hard-hearted of prel- 
ates, and the defence is that he took his 
little son on his knee and kissed him; we 
censure him for having violated the arti- 
cles of the Petition of Rights, after having 
for a good and valuable consideration 
promised to observe them, and we are 
informed that he was accustomed to hear 
prayers at six o'clock in the morning. 
What fallacy does Macaulay refer to in this passage ? 



PRACTICAL EXERCISES AND QUESTIONS 24 1 

(21) This must be a bad measure, because it is 

supported by bad men. 

(22) For those who are bent on cultivating their 

minds by dihgent study, the incitement 
of academic honors is unnecessary; and 
it is ineffectual for the idle, and such as are 
indifferent to mental improvement ; there- 
fore, the incitement of academic honors 
should be abolished. 

(23) Logic as it was cultivated by the Schoolmen 

was a useless study; therefore, logic as it 
is cultivated to-day is a fruitless study. 

(24) Protective laws should be abohshed; for 

they are injurious if they produce scarcity, 
and they are useless if they do not. 

(25) What is the good of all your teaching, for 

every day we hear of forgeries, which 
would never have been committed by 
these persons, had they not learned to 
read and write ? 

(26) Does a grain of millet, when dropped on the 

floor, make a sound ? No. Does a bushel 
of millet under these same circumstances 
make a sound? Yes. Is there not a 
determinate proportion between the bushel 
and the grain? There is. There must, 
therefore, be the same proportion between 
the sonorousness of the two. If one 
grain be not sonorous, neither can ten 
thousand grains be so. 



242 ELEMENTARY LOGIC 

(27) He that can swim need not despair to fly; 

for to swim is to fly in a grosser fluid, and 
to fly is to swim in a subtler fluid. 

(28) The more correct the logic, the more cer- 

tainly will the conclusion be wrong, if the 
premises are false; therefore, where the 
premises are wholly uncertain the best 
logician is the least safe guide. 

(29) The two propositions, Aristotle is living and 

Aristotle is dead, are both intelHgible prop- 
ositions; they are both of them true or 
both of them false, because all intelHgible 
propositions must be either true or false. 

(30) Every incident in the narration is probable; 

hence the narrative is probable. 

(31) The end of a thing is its perfection; death 

is the end of life, and therefore death is 
the perfection of life. 

(32) It is enough to reply to your argument, when 

I remind you that of all men you should 
be last to advocate this doctrine. 

(33) Every law is either useless or it occasions 

hurt to some person; now, a law that is 
useless ought to be abolished, and so 
ought a law that occasions harm; there- 
fore, every law ought to be abolished. 

(34) All the plays of Shakespeare cannot be read 

in a day; "Hamlet "is a play of Shake- 
speare, and consequently it cannot be 
read in a day. 



PRACTICAL EXERCISES AND QUESTIONS 243 

(35) You are not what I am; I am a man; there- 

fore, you are not a man. 

(36) Theft is a crime; theft was encouraged by 

the laws of Sparta; therefore, the laws 
of Sparta encouraged crime. 

(37) The Greeks produced masterpieces of art; 

the Spartans were Greeks, and therefore 
they produced masterpieces of art. 

(38) He is the greatest lover of any one who seeks 

that person's greatest good; a virtuous 
man seeks the greatest good for himself; 
therefore the virtuous man loves himself 
most. 

(39) The student of history is compelled to ad- 

mit the law of progress, for he finds that 
society has never stood still. 

EXAMPLES OF THE METHODS OF OBSERVATION AND 
EXPERIMENT IN ASCERTAINING CAUSAL CONNECTION 

I. Baron Liebeg investigated the actions of certain 
metalHc poisons. His problem was to ascertain the 
property common to arsenious acid, salts of lead, bis- 
muth, copper, and mercury on which their destructive 
action was dependent. He ascertained the following 
facts : — 

(i) When solutions of these substances are placed 
in sufficiently close contact with many ani- 
mal products, albumen, milk, muscular 
fiber, and animal membranes, the acid or 
salt leaves the water in which it was dis- 



244 ELEMENTARY LOGIC 

solved, and enters into combination with 
the animal substances; which substance, 
after being acted upon, loses its tendency to 
spontaneous decomposition or putrefaction. 

(2) In all cases where death has been produced by 

these poisons, those parts of the body with 
which these substances have been brought 
into contact do not afterwards putrefy. 

(3) When too small a quantity of poison has been 

used to destroy hfe, eschars are produced, 
superficial portions of the tissues are de- 
stroyed. 

(4) Many insoluble basic salts are known not to 

be poisonous; when these, however, are 
brought into contact with tissues, they do 
not combine with them so as to arrest the 
process of decomposition. 

(5) Antidotes to these poisons are known to com- 

bine with them so as to prevent them from 
acting upon the tissues, by forming insolu- 
ble compounds. 
The conclusion reached by Baron Liebeg was that 
the proximate cause of death from the action of these 
poisons is the conversion of animal tissues into a chemi- 
cal compound, held together by so powerful a force as 
to resist the action of the ordinary causes of decompo- 
sition on which the continuance of life depends. 
By what method was this conclusion reached? 
2. Dr. Wells' investigation upon the cause of noc- 
turnal dew: — 



PRACTICAL EXERCISES AND QUESTIONS 245 

The following facts were ascertained : — 

(i) Whenever an object contracts dew it is colder 
than the air. 

(2) No dew is produced on the surface of polished 

metals, but dew is produced on the surface 
of glass. 

(3) Polished substances which conduct heat least 

were found to be most conspicuously be- 
dewed, while those which conduct it will 
resist dew. 

(4) Substances which part with their heat most 

readily by radiation contract dew most 
abundantly. 

(5) All the instances in which much dew is de- 

posited agree in this circumstance only, 
that they either radiate heat rapidly, or 
conduct it slowly. Bodies that are be- 
dewed arc those that lose heat from the 
surface faster than it is restored from 
within. 

(6) Dew is never deposited copiously in situa- 

tions much screened from the open sky 
and not at all in a cloudy night. 
Which of the methods were followed in ascertaining 
these facts, and in reaching the conclusion in 5 ? 

3. What method was followed by Arfwedson in his 
discovery of lithia by noting an excess of weight in the 
sulphate produced from a small portion of what he con- 
sidered as magnesium present in a mineral he had ana- 
lyzed ? 



246 ELEMENTARY LOGIC 

4. Jevons observed that economic crises have oc- 
curred at regular intervals of about ten years ; this ten 
years' periodicity, moreover, seems to correspond to a 
similar periodicity of bad harvests; and the causes of 
this seem to be a decennial periodicity of spots on the 
sun. 

5. In a simple fracture of the ribs if the lung be punc- 
tured by a fragment, the blood effused into the pleural 
cavity, although freely mixed with air, undergoes no 
decomposition. That is not the case if air enter directly 
through a wound in the chest. This difference in re- 
sult must be causally connected with special circum- 
stances — viz. passage of air through tissues in the lungs. 

What method is illustrated in these observations? 

6. If the lung be emptied as perfectly as possible, 
and a handful of cotton wool be placed against the 
mouth and nostrils, and you inhale through it, it will 
be found on expiring this air through a glass tube that 
its freedom from floating matter is manifest. 

What two circumstances are shown to be causally 
connected in this experiment and by what method ? 

7. The following experiments, it is maintained, prove 
that the feeling of effort is of peripheral rather than 
central origin. 

(i) Hold the finger as if to pull a trigger; think 
vigorously of bending it but do not bend it ; 
an unmistakable feeling of effort results. 
Note in repeating this experiment, that the 
breath is involuntarily held, and that there 
are also other muscle contractions. 



PRACTICAL EXERCISES AND QUESTIONS 247 

(2) Now, repeat the experiment, and breathe 
regularly at the same time and avoid other 
muscle contractions, and note that no feel- 
ing of effort is the result. What method is 
followed in these experiments ? 
8. Darwin asserted that cross f ertihzation of the flower 
of the common broom by bees is causally connected 
with a curious mechanism in these flowers. The fol- 
lowing circumstances were noted by him : — 

(i) "When a bee ahghts on the petals of a young 
flower it is shghtly opened, and short sta- 
mens spring out, which rub their pollen 
against the abdomen of the bee. If a 
rather older flower is visited for the first 
time (or if the bee exerts great force on a 
younger flower), the keel opens along its 
whole length, and the longer as well as the 
shorter stamens, together with the much 
elongated curved pistil, spring forth with 
violence. The flattened spoonlike extrem- 
ity of the pistil rests for a time on the back 
of the bee, and leaves on it the load of pollen 
with which it is charged. As soon as the 
bee flies away, the pistil instantly curls 
round, so that the stigmatic surface is now 
upturned and occupies a position in which 
it would be rubbed against the abdomen of 
another bee visiting the same flower. Thus, 
when the pistil first escapes from the keel, 
the stigma is rubbed against the back of 



248 ELEMENTARY LOGIC 

the bee, dusted with pollen from the shorter 
stamens, which is often shed a day or two 
before that from the longer stamens. If 
the visits of bees are prevented, and if the 
flowers are not dashed by the wind against 
any object, the keel never opens, so that the 
stamens and pistil remain inclosed. Plants 
thus protected yield very few pods in com- 
parison with those produced by neighbor- 
ing uncovered bushes, and sometimes none 
at all." Quoted from Darwin in Hibbens's 
"Inductive Logic," pp. 316-317. 

9. Kenelm Digby's treatment of wounds was to 
apply an ointment, not to the wound itself, but to the 
sword that had inflicted it, to dress this carefully at 
regular intervals, and, in the meantime having bound up 
the wound, to leave it alone for seven days. It was 
observed that many cures followed upon this treatment. 

What fallacies does this incident illustrate ? 

10. What fallacy underlies the saying "Fortune 
favors fools"? 

11. By what fallacious methods is the success of 
patent medicines largely promoted ? 

12. To what were the following beliefs chiefly ow- 
ing? 

A body ten times as heavy as another falls ten 
times as fast. Objects immersed in water are 
always magnified. The magnet exerts an 
irresistible force. Crystals are always found 
associated with ice. 



PRACTICAL EXERCISES AND QUESTIONS 249 

13. A belief was current in Adam Smith's time that 
prodigaHty encourages industry and parsimony dis- 
courages it. Observation seemed to justify this behef ; 
those who spent lavishly gave great employment to 
labor. Those who were not thus lavish did not appear 
to do so. 

What faults of observation were the cause of this 
erroneous behef? 

14. What mal-observation was there in the objection 
to free trade, that the purchase of British silk encour- 
ages British industry, the purchase of Lyons silk 
encourages only French industry ? 

15. What error hes in the following behefs? What- 
soever has never been will never be. Women as a 
class are not equal to men. Society cannot prosper 
without slavery. Philosophers are impractical men. 

16. What fallacy can you charge against the follow- 
ing arguments? 

''As there could be in natural bodies no motion 
of anything unless there were some which 
moveth all things, and continueth immovable ; 
even so in pohtic societies there must be some 
unpunishable, or else no man shall suffer 
punishment." 

"It would be admitted that a great and perma- 
nent diminution in the quantity of some useful 
commodity, such as corn, or coal, or iron 
throughout the world, would be a serious and 
lasting loss; and again, that if the fields and 
coal mines yielded regularly double quantities, 



250 ELEMENTARY LOGIC 

with the same labor, we should be so much 
the richer; hence it might be inferred, that if 
the quantity of gold and silver in the world 
were diminished one half, or were doubled, 
like results would follow; the utiHty of these 
metals for the purpose of coin being very 
great." 



ARGUMENTATION AND DEBATE 

By 
CRAVEN LAYCOCK 

Assistant Professor of Oratory in Dartmouth College 

and 
ROBERT LEIGHTON SCALES 

Instructor in English in Dartmouth College 

i2mo Cloth zviii + 361 pages $1.10 net 



In this work the peculiar difficulties which stand in the way of making 
a practical text-book for use in teaching argumentation and debate 
have been overcome. The authors have succeeded in producing a 
book which is not only practical and teachable, but which has the still 
rarer quality of being easily understood. The treatment of the topics 
presented — the proposition, the issues, preliminary reading, evidence, 
kinds of arguments, fallacies, brief-drawing, the principles of presenta- 
tion, refutation, and debate — is lucid and interesting as well as highly 
profitable. 

The discussion of the Issues is built around the vital statement that, 
" in arguing, there are always certain ideas or matters of fact, upon the 
establishment of which depends the establishment of the proposition." 
It is shown that there is no sure way of guarding against irrelevant 
discussion, except by clear understanding and concise statement of the 
issues. The method of finding the issues is fully explained and is also 
illustrated by quotations from the speeches of great debaters. 

The discussion of Evidence rests on the broad legal basis. Under 
this head are included a careful analysis of the kinds of evidence, and 
a discriminating statement of the relative value of evidence, while the 
various tests of evidence are suggested for the consideration of the 
student. At the close of the chapter on Evidence will be found an 
illustration of the method employed, which makes the book eminently 
teachable. The whole chapter is summarized in such a way that, not 
only is the student greatly assisted in fixing in his memory the various 
topics, but the instructor likewise finds great help in effective quizzing. 



The difference between Evidence and Arguments is made clear in 
the chapter on Kinds of Arguments. It is strongly maintained that 
the different kinds of Arguments may all be best explained by direct 
reference to the causal connection between that which is known and 
that which is to be inferred. In every case the discussion is strength- 
ened by well-chosen illustrations from standard sources. 

Perhaps the most distinctive chapter in the book is that on Brief- 
drawing. This subject is so presented that even the beginner, after 
careful study of the chapter, is ready to commence the work for him- 
self. The plan of the chapter is as follows : {a) A subject in the form 
of a proposition is selected ; {p) a rough outline is made ; (<r) this 
rough outline is modified step by step ; (^) there are discussions of 
each change and the formulation of a rule of brief-drawing, and 
finally (^) the completed brief is given as a whole. At the end of the 
chapter are provided all the rules that have been enunciated for the 
drawing of briefs. This method of presenting this most difficult part of 
the work is, it is believed, vastly superior to any other method that has 
been tried, in that it gives the student one model, drawn on approved 
lines and presented clearly, thus guarding against what too often 
becomes a discouraging jumble. 

For apt Illustrations, that actually illuminate the processes under 
discussion, the great preachers, platform speakers, and forensic orators 
of ancient and modern times, have been so laid under contribution, 
that the result is practically an anthology of argumentation and debate. 

In the work under consideration it is fully realized that something ■ 
more is needed for Debate than for written discussion, and so the Part 
on Debate contains added suggestions of a highly practical nature. 

Teachers will find that the careful paragraphing, leiterittg, and sum- 
marizing, which have been done with great exactness throughout the 
entire book, help in no small measure to make the subject of Argu- 
mentation and Debate eminently practical for class use. The Ap- 
pendixes contain suggestions of additional exercises that have proved 
helpful in the teaching of the subject. 



THE MACMILLAN COMPANY 

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